Almost Surely Investigations in Optimal Control and Dynamical Systems,by Wouter Jongeneel.
Hi, in January 2020 I started as a EE PhD student at EPFL, as part of the NCCR Automation, in the Risk Analytics & Optimization group lead by Daniel Kuhn. Quote of the month (October 2023) Weekend read May 14, 2023

Two disjoint closed sets that are at a zero distance.
This example is not that surprising, but it nicely illustrates the pitfall of thinking of sets as ‘‘blobs’’ (in Dutch we would say potatoes) in the plane. Indeed, the counterexample can be interpreted as an asymptotic result. Consider , this set is closed as is continuous. Then, take , this set is again closed and clearly and are disjoint. However, since we can write , then as , we find that approaches arbitrarily closely. One can create many more examples of this form, e.g., using logarithms.
An open and closed metric ball with the same center and radius such that the closure of the open ball is unequal to the closed ball.
This last example is again not that surprising if you are used to go beyond Euclidean spaces. Let be a metric space under the metric
Now assume that is not simply a singleton. Then, let be the open metric ball of radius centered at and similarly, let be the closed metric ball of radius centered at . Indeed, whereas . Now to construct the closure of , we must first understand the topology induced by the metric . In other words, we must understand the limit points of . Indeed, is the discrete metric and this metric induces the discrete topology on . Hence, it follows — which is perhaps the counterintuitive part, that and hence . This example also nicely shows a difference between metrics and norms. As norms satisfy (absolute homogeneity) one cannot have the discontinuous behaviour from the example above.
Further examples I particularly enjoyed are a discontinuous linear function, a function that is discontinuous in two variables, yet continuous in both variables separately and all examples related to Cantor. This post is inspired by TAing for Analysis 1 and will be continued in the near future.
In the last issue of the IEEE Control Systems Magazine, Prof. Sepulchre wrote an exhilarating short piece on the art of the state, or as he (also) puts it ‘‘the definite victory of computation over modeling in science and engineering’’. As someone with virtually no track record, I am happy to also shed my light on the matter.
We owe dynamical systems theory as we know it today largely to Henri Poincaré (18541912), whom started this in part being motivated by describing the solar system. Better yet, the goal was to eventually understand its stability. However, we can be a bit more precise regarding what was driving all this work. In the introduction of Les méthodes nouvelles de la mécanique céleste (EN: The new methods of celestial mechanics)  Vol I by Poincaré, published in 1892 (HP92), we find (loosely translated): ‘‘The ultimate goal of Celestial Mechanics is to resolve this great question of whether Newton's law alone explains all astronomical phenomena; the only way to achieve this is to make observations as precise as possible and then compare them with the results of the calculation.’’ and then after a few words on divergent series: ‘‘these developments should attract the attention of geometers, first for the reasons I have just explained and also for the following: the goal of Celestial Mechanics is not reached when one has calculated more or less approximate ephemerides without being able to realize the degree of approximation obtained. If we notice, in fact, a discrepancy between these ephemerides and the observations, we must be able to recognize if Newton's law is at fault or if everything can be explained by the imperfection of the theory.’’ It is perhaps interesting to note that his 1879 thesis entitled: Sur les propriétés des fonctions définies par les équations aux différences partielles (EN: On the properties of functions defined by partial difference equations) was said to lack examples (p.109 FV12).
In the 3 Volumes and well over a 1000 pages of the new celestial mechanics Poincaré develops a host of tools towards the goal sketched above. Indeed, this line of work contributed to the popularizing of the infamous equation
or as used in control theory: , for some ‘‘update’’ operator. However, I would like to argue that the main issue with Equation is not that it is frequently used, but how it is used, moving from the qualitative to the quantitative.
Unfortunately we cannot ask him, but I believe Poincaré would wholeheartedly agree with Sepulchre. Inspired by the beautiful book by Prof. Verhulst (FV12) I will try to briefly illustrate this claim. As put by Verhulst (p.80 FV12): ‘‘In his writings, Poincaré was interested primarily in obtaining insight into physical reality and mathematical reality — two different things.’’ In fact, the relation between analysis and physics was also the subject of Poincaré’s plenary lecture at the first International Congress of Mathematicians, in Zurich (1897). (p.91 FV12).
In what follows we will have a closer look at La Valeur de la Science (EN: The value of science) by Poincaré, published in 1905 (HP05). This is one of his more philosophical works. We already saw how Poincaré was largely motivated by a better understanding of physics. We can further illustrate how he — being one of the founders of many abstract mathematical fields — thought of physics. To that end, consider the following two quotes:
’’when physicists ask of us the solution of a problem, it is not a dutyservice they impose upon us, it is on the contrary we who owe them thanks.’’ (p.82 HP05)
‘‘Fourier's series is a precious instrument of which analysis makes continual use, it is by this means that it has been able to represent discontinuous functions; Fourier invented it to solve a problem of physics relative to the propagation of heat. If this problem had not come up naturally, we should never have dared to give discontinuity its rights; we should still long have regarded continuous functions as the only true functions.’’ (p.81 HP05)
What is more, this monograph contains a variety of discussions of the form: ’’If we assume A, B and C, what does this imply?’’ Both mathematically and physically. For instance, after a discussion on Carnot's Theorem (principle) — or if you like the ramifications of a global flow — Poincaré writes the following:
‘‘On this account, if a physical phenomenon is possible, the inverse phenomenon should be equally so, and one should be able to reascend the course of time. Now, it is not so in nature, …’’ (p.96 HP05)
However, Poincaré continues and argues that although notions like friction seem to lead to an irreversible process this is only so because we look at the process through the wrong lens.
‘‘For those who take this point of view, Carnot's principle is only an imperfect principle, a sort of concession to the infirmity of our senses; it is because our eyes are too gross that we do not distinguish the elements of the blend; it is because our hands are too gross that we can not force them to separate; the imaginary demon of Maxwell, who is able to sort the molecules one by one, could well constrain the world to return backward.’’ (p.97 HP05)
Indeed, the section is concluded by stating that perhaps we should look through the lens of statistical mechanics. All to say that if the physical reality is ought the be described, this must be on the basis of solid principles. Let me highlight another principle: conservation of energy. Poincaré writes the following:
‘‘I do not say: Science is useful, because it teaches us to construct machines. I say: Machines are useful, because in working for us, they will some day leave us more time to make science.’’ (p.88 HP05)
‘‘Well, in regard to the universe, the principle of the conservation of energy is able to render us the same service. The universe is also a machine, much more complicated than all those of industry, of which almost all the parts are profoundly hidden from us; but in observing the motion of those that we can see, we are able, by the aid of this principle, to draw conclusions which remain true whatever may be the details of the invisible mechanism which animates them.’’ (p.94 HP05)
Then, a few pages later, his brief story on radium starts as follows:
‘‘At least, the principle of the conservation of energy yet remained to us, and this seemed more solid. Shall I recall to you how it was in its turn thrown into discredit?’’ (p.104 HP05)
Poincaré his viewpoint is to some extent nicely captured by the following piece in the book by Verhulst:
‘‘Consider a mature part of physics, classical mechanics (Poincaré 02). It is of interest that there is a difference in style between British and continental physicists. The former consider mechanics an experimental science, while the physicists in continental Europe formulate classical mechanics as a deductive science based on a priori hypotheses. Poincaré agreed with the English viewpoint, and observed that in the continental books on mechanics, it is never made clear which notions are provided by experience, which by mathematical reasoning, which by conventions and hypotheses.’’ (p.89 FV12)
In fact, for both mathematics and physics Poincaré spoke of conventions, to be understood as a convenience in physics. Overall, we see that Poincaré his attitude to applying mathematical tools to practical problems is significantly more critical than what we see nowadays in some communities. For instance:
‘‘Should we simply deduce all the consequences, and regard them as intangible realities? Far from it; what they should teach us above all is what can and what should be changed.’’ (p.79 HP05)
A clear example of this viewpoint appears in his 1909 lecture series at Göttingen on relativity:
‘‘Let us return to the old mechanics that admitted the principle of relativity. Instead of being founded on experiment, its laws were derived from this fundamental principle. These considerations were satisfactory for purely mechanical phenomena, but the old mechanics didn’t work for important parts of physics, for instance optics.’’ (p.209 FV12)
Now it is tempting to believe that Poincaré would have liked to get rid of ‘‘the old’’, but this is far from the truth.
‘‘We should not have to regret having believed in the principles, and even, since velocities too great for the old formulas would always be only exceptional, the surest way in practise would be still to act as if we continued to believe in them. They are so useful, it would be necessary to keep a place for them. To determine to exclude them altogether would be to deprive oneself of a precious weapon.’’ (p.111 HP05)
The previous was about the ‘‘old’’ and ‘‘new’’ mechanics and how differences in mathematical and physical realities might emerge. Most and for all, the purpose of the above is to show that one of the main figures in abstract dynamical system theory — someone that partially found the field motivated by a computational question — greatly cared about correct modelling.
This is one part of the story. The second part entails the recollection that Poincaré his work was largely qualitative. Precisely this qualitative viewpoint allows for imperfections in the modelling process. However, this also means conclusions can only be qualitative. To provide two examples; based on the HartmanGrobman theorem and the genericity of hyperbolic equilibrium points it is perfectly reasonable to study linear dynamical systems, however, only to understand local qualitative behaviour of the underlying nonlinear dynamical system. The aim to draw quantitative conclusions is often futile and simply not the purpose of the tools employed. Perhaps more interesting and closer to Poincaré, the Lorenz system, being a simplified mathematical model for atmospheric convection, is wellknown for being chaotic. Now, weatherforecasters do use this model, not to predict the weather, but to elucidate why their job is hard!
Summarizing, I believe the editorial note by Prof. Sepulchre is very timely as some have forgotten that the initial dynamical systems work by Poincaré and others was to get a better understanding, to adress questions, to learn from, not to end with. Equation (1) is not magical, but a tool to build the bridge, a tool to see if and where the bridge can be built.
Recently, DeepMind published new work on ‘‘discovering’’ numerical linear algebra routines. This is remarkable, but I leave the assesment of this contribution to others. Rather, I like to highlight the earlier discovery of Strassen. Namely, how to improve upon the obvious matrix multiplication algorithm.
To get some intuition, consider multiplying two complex numbers , that is, construct . In this case, the obvious thing to do would lead to 4 multiplications, namely , , and . However, consider using only , and . These multiplications suffice to construct (subtract the latter two from the first one). The emphasis is on multiplications as it seems to be folklore knowledge that multiplications (and especially divisions) are the most costly. However, note that we do use more additive operations. Are these operations not equally ‘‘costly’’? The answer to this question is delicate and still changing!
When working with integers, additions are indeed generally ‘‘cheaper’’ than multiplications. However, oftentimes we work with floating points. Roughly speaking, these are numbers characterized by some (signed) mantissa , a base and some (signed) exponent , specifically, . For instance, consider the addition . To perform the addition, one normalizes the two numbers, e.g., one writes . Then, to perform the addition one needs to take the appropriate precision into account, indeed, the result is . All this example should convey is that we have sequential steps. Now consider multiplying the same numbers, that is, . In this case, one can proceed with two steps in parallel, on the one hand and on the other hand . Again, one needs to take the correct precision into account in the end, but we see that by no means multiplication must be significantly slower in general!
Now back to the matrix multiplication algorithms. Say we have
then, is given by
Indeed, the most straightforward matrix multiplication algorithm for and costs you multiplications and additions. Using the intuition from the complex multiplication we might expect we can do with fewer multiplications. In 1969 (Numer. Math. 13 354–356) Volker Strassen showed that this is indeed true.
Following his short, but highly influential, paper, let , , , , , and . Then, , , and . See that we need multiplications and additions to construct for both and being dimensional. The ‘‘obvious’’ algorithm would cost multiplications and additions. Indeed, one can show now that for matrices of size one would need multiplications (via induction). Differently put, one needs multiplications. This bound prevails when including all operations, but only asymptotically, in the sense that . In practical algorithms, not only multiplications, but also memory and indeed additions play a big role. I am particularly interested how the upcoming DeepMind algorithms will take numerical stability into account.
*Update: the improved complexity got immediately improved.
The lack of mathematically oriented people in the government is an often observed phenomenon 1. In this short note we clarify that this is a corollary to ‘‘An application of Poincare's recurrence theorem to adademic adminstration’’, by Kenneth R. Meyer. In particular, since the governmental system is qualitatively the same as that of academic administration, it is also conservative. However  due to wellknown results also pioneered by Poincare  this implies the following.
Corollary: The government is nonexact.
A while ago Prof. Jan H. van Schuppen published his book Control and System Theory of DiscreteTime Stochastic Systems. In this post I would like to highlight one particular concept from the book: (stochastic) coobservability, which is otherwise rarely discussed.
We start with recalling observability. Given a lineartimeinvariant system with , :
one might wonder if the initial state can be recovered from a sequence of outputs . (This is of great use in feedback problems.) By observing that , , one is drawn to the observability matrix
Without going into detectability, when is fullrank, one can (uniquely) recover (proof by contradiction). If this is the case, we say that , or equivalenty the pair , is observable. Note that using a larger matrix (more data) is redudant by the CayleyHamilton theorem (If would not be fullrank, but by adding “from below” it would somehow be fullrank, one would contradict the CayleyHamilton theorem.). Also note that in practice one does not “invert” but rather uses a (Luenberger) observer (or a Kalman filter).
Now lets continue with a stochastic (Gaussian) system, can we do something similar? Here it will be even more important to only think about observability matrices as merely tools to assert observability. Let be a zeromean Gaussian random variable with covariance and define for some matrices and the stochastic system
We will assume that is asymptotically (exponentially stable) such that the Lyapunov equation describing the (invariant) statecovariance is defined:
Now the support of the state is the range of .
A convenient tool to analyze will be the characteristic function of a Gaussian random variable , defined as . It can be shown that for a Gaussian random variable . With this notation fixed, we say that is stochastically observable on the internal if the map
is injective on the support of (note the ). The intuition is the same as before, but now we want the state to give rise to an unique (conditional) distribution. At this point is seems rather complicated, but as it turns out, the conditions will be similar to ones from before. We start by writing down explicit expressions for , as
we find that
for the observability matrix corresponding to the data (length) of , a matrix containing all the noise related terms and a stacked vector of noises similar to . It follows that is given by , for . Injectivity of this map clearly relates directly to injectivity of . As such (taking the support into account), a neat characterization of stochastic observability is that .
Then, to introduce the notion of stochastic coobservability we need to introduce the backward representation of a system. We term system representations like and “forward” representations as . Assume that , then see that the forward representation of a system matrix, denoted is given by . In a similar vein, the backward representation is given by . Doing the same for the output matrix yields and thereby the complete backward system
Note, to keep and fixed, we adjust the distribution of . Indeed, when is not fullrank, the translation between forward and backward representations is not welldefined. Initial conditions cannot be recovered.
To introduce coobservability, ignore the noise for the moment and observe that , , and so forth. We see that when looking at observability using the backward representation, we can ask if it possible to recover the current state using past outputs. Standard observability looks at past states instead. With this in mind we can define stochastic coobservability on the interval be demanding that the map
is injective on the support of (note the ). Of course, one needs to make sure that is defined. It is no surprise that the conditions for stochastic coobservability will also be similar, but now using the coobservability matrix. What is however remarkable, is that these notions do not always coincide.
Lets look at when this can happen and what this implies. One reason to study these kind of questions is to say something about (minimal) realizations of stochastic processes. Simply put, what is the smallest (as measured by the dimension of the state ) system that gives rise to a certain output process. When observability and coobservability do not agree, this indicates that the representation is not minimal. To get some intuition, we can do an example as adapted from the book. Consider the scalar (forward) Gaussian system
for . The system is stochastically observable as and . Now for stochastic coobservability we see that , as such the system is not coobservable. What this shows is that behaves as a Gaussian random variable, no internal dynamics are at play and such a minimal realization is of dimension .
For this and a lot more, have a look at the book!
When learning about convex sets, the definitions seem so clean that perhaps you think all is known what could be known about finitedimensional convex geometry. In this short note we will look at a problem which is still largely open beyond the planar case. This problem is called the fair partitioning problem.
In the dimensional case, the question is the following, given any integer , can a convex set be divided into convex sets all of equal area and perimeter. Differently put, does there exist a fair convex partitioning, see Figure 1.
Figure 1: A partition of into fair convex sets. 
This problem was affirmatively solved in 2018, see this paper.
As you can see, this work was updated just a few months ago. The general proof is involved, lets see if we can do the case for a compact set and .
First of all, when splitting (into just two sets) you might think of many different methods to do so. What happens when the line is curved? The answer is that when the line is curved, one of the resulting sets must be nonconvex, compare the options in Figure 2.
Figure 2: A partitioning of the square in into two convex sets can only be done via a straight cut. 
This observation is particularly useful as it implies we only need to look at two points on the boundary of (and the line between them). As is compact we can always select a cut such that the resulting perimeters of and are equal.
Let us assume that the points and in Figure 3.(i) are like that. If we start moving them around with equal speed, the resulting perimeters remain fixed. Better yet, as the cut is a straightline, the volumes (area) of the resulting set and change continuously. Now the result follows from the Intermediate Value Theorem and seeing that we can flip the meaning of and , see Figure 3.(iii).
Figure 3: By moving the points and we continuously change and . 
In this post we will look at one of the many remarkable findings by Roger W. Brockett. Consider a Linear Program (LP)
parametrized by the compact set and a suitable triple . As a solution to can always be found to be a vertex of , a smooth method to solve seems somewhat awkward. We will see that one can construct a socalled isospectral flow that does the job. Here we will follow Dynamical systems that sort lists, diagonalize matrices and solve linear programming problems, by Roger. W. Brockett (CDC 1988) and the book Optimization and Dynamical Systems edited by Uwe Helmke and John B. Moore (Springer 2ed. 1996). Let have vertices, then one can always find a map , mapping the simplex onto . Indeed, with some abuse of notation, let be a matrix defined as , for , the vertices of .
Before we continue, we need to establish some differential geometric results. Given the Special Orthogonal group , the tangent space is given by . Note, this is the explicit formulation, which is indeed equivalent to shifting the corresponding Lie Algebra. The easiest way to compute this is to look at the kernel of the map defining the underlying manifold.
Now, following Brockett, consider the function defined by for some . This approach is not needed for the full construction, but it allows for more intuition and more computations. To construct the corresponding gradient flow, recall that the (Riemannian) gradient at is defined via for all . Using the explicit tangent space representation, we know that with . Then, see that by using
we obtain the gradient via
This means that (the minus is missing in the paper) the (a) gradient flow becomes
Consider the standard commutator bracket and see that for one obtains from the equation above (typo in the paper)
Hence, can be seen as a reparametrization of a gradient flow. It turns out that has a variety of remarkable properties. First of all, see that preserves the eigenvalues of . Also, observe the relation between extremizing and the function defined via . The idea to handle LPs is now that the limiting will relate to putting weight one the correct vertex to get the optimizer, gives you this weight as it will contain the corresponding largest costs.
In fact, the matrix can be seen as an element of the set . This set is in fact a smooth compact manifold as it can be written as the orbit space corresponding to the group action , , one can check that this map satisfies the group properties. Hence, to extremize over , it appears to be appealing to look at Riemannian optimization tools indeed. When doing so, it is convenient to understand the tangent space of . Consider the map defining the manifold , . Then by the construction of , see that yields the relation for any .
For the moment, let such that and . First we consider the convergence of . Let have only distinct eigenvalues, then exists and is diagonal. Using the objective from before, consider and see that by using the skewsymmetry one recovers the following
This means the cost monotonically increases, but by compactness converges to some point . By construction, this point must satisfy . As has distinct eigenvalues, this can only be true if itself is diagonal (due to the distinct eigenvalues).
More can be said about , let be the eigenvalues of , that is, they correspond to the eigenvalues of as defined above. Then as preserves the eigenvalues of (), we must have , for a permutation matrix. This is also tells us there is just a finite number of equilibrium points (finite number of permutations). We will write this sometimes as .
Now as is one of those points, when does converge to ? To start this investigation, we look at the linearization of , which at an equilibrium point becomes
for . As we work with matrixvalued vector fields, this might seems like a duanting computation. However, at equilibrium points one does not need a connection and can again use the directional derivative approach, in combination with the construction of , to figure out the linearization. The beauty is that from there one can see that is the only asymptotically stable equilibrium point of . Differently put, almost all initial conditions will converge to with the rate captured by spectral gaps in and . If does not have distinct eigenvalues and we do not impose any eigenvalue ordering on , one sees that an asymptotically stable equilibrium point must have the same eigenvalue ordering as . This is the sorting property of the isospectral flow and this is of use for the next and final statement.
Theorem: Consider the LP with for all , then, there exist diagonal matrices and such that converges for almost any to with the optimizer of being .
Proof: Global convergence is prohibited by the topology of . Let and let . Then, the isospectral flow will converge from almost everywhere to (only ), such that .
Please consider the references for more on the fascinating structure of .
Most of us will learn at some point in our life that is problematic, as which number multiplied by itself can ever be negative? To overcome this seemingly useless defiency one learns about complex numbers and specifically, the imaginary number , which is defined to satisfy . At this point you should have asked yourself when on earth is this useful?
In this short post I hope to highlight  from a slightly different angle most people grounded in physics would expect  that the complex numbers are remarkably useful.
Complex numbers, and especially the complex exponential, show up in a variety of contexts, from signal processing to statistics and quantum mechanics. With of course most notably, the Fourier transformation.
We will however look at something completely different. It can be argued that in the late 60s James Lyness and Cleve Moler brought to life a very elegant new approach to numerical differentiation. To introduce this idea, recall that even nowadays the most wellknown approach in numerical differentiation is to use some sort of finitedifference method, for example, for any one could use the centraldifference method
Now one might be tempted to make extremely small, as then the error must vanish! However, numerically, for a very small the two function evaluations and will be indistinguishable. So although the error scales as there is some practical lower bound on this error based on the machine precision of your computer. One potential application of numerical derivatives is in the context of zerothorder (derivativefree) optimization. Say you want to adjust van der Poel his Canyon frame such that he goes even faster, you will not have access to explicit gradients, but you can evaluate the performance of a change in design, for example in a simulator. So what you usually can obtain is a set of function evaluations . Given this data, a somewhat obvious approach is to mimick firstorder algorithms
where is some stepsize. For example, one could replace in by the centraldifference approximation . Clearly, if the objective function is wellbehaved and the approximation of is reasonably good, then something must come out? As was remarked before, if your approximation  for example due to numerical cancellation errors  will always have a bias it is not immediate how to construct a highperformance zerothorder optimization algorithm. Only if there was a way to have a numerical approximation without finite differencing?
Let us assume that is sufficiently smooth, let be the imaginary number and consider the following series
From this expression it follows that
So we see that by passing to the complex domain and projecting the imaginary part back, we obtain a numerical method to construct approximations of derivatives without even the possibility of cancellation errors. This remarkable property makes it a very attractive candidate to be used in zerothorder optimization algorithms, which is precisely what we investigated in our new preprint. It turns out that convergence is not only robust, but also very fast!
In this post we discuss very classical, yet, highly underappreciated topological results.
We will look at dynamical control systems of the form
where is some finitedimensional embedded manifold.
As it turns out, topological properties of encode a great deal of fundamental control possibilities, most notably, if a continuous globally asymptotically stabilizing control law exists or not (usually, one can only show the negative). In this post we will work towards two things, first, we show that a technically interesting nonlinear setting is that of a dynamical system evolving on a compact manifold, where we will discuss that the desirable continuous feedback law can never exist, yet, the boundedness implied by the compactness does allow for some computations. Secondly, if one is determined to work on systems for which a continuous globally asymptotically stable feedback law exists, then a wide variety of existential sufficient conditions can be found, yet, the setting is effectively Euclidean, which is the most basic nonlinear setting.
To start the discussion, we need one important topological notion.
Definition [Contractability]. Given a topological manifold . If the map is nullhomotopic, then, is contractible.
To say that a space is contractible when it has no holes (genus ) is wrong, take for example the sphere. The converse is true, any finitedimensional space with a hole cannot be contractible. See Figure 1 below.
Figure 1: The manifold is contractible, but and are not. 
In this section we highlight a few (not all) of the most elegant topological results in stability theory. We focus on results without appealing to the given or partially known vector field . This line of work finds its roots in work by Bhatia, Szegö, Wilson, Sontag and many others.
Theorem [The domain of attraction is contractible, Theorem 21] Let the flow be continuous and let be an asymptotically stable equilibrium point. Then, the domain of attraction is contractible.
This theorem by Sontag states that the underlying topology of a dynamical system, better yet, the space on which the dynamical system evolves, dictates what is possible. Recall that linear manifolds are simply hyperplanes, which are contractible and hence, as we all know, global asymptotic stability is possible for linear systems. However, if is not contractible, there does not exist a globally asymptotically stabilizable continuoustime system on , take for example any sphere.
The next example shows that one should not underestimate ‘‘linear systems’’ either.
Example [Dynamical Systems on ] Recall that is a smooth dimensional manifold (Lie group). Then, consider for some , the (rightinvariant) system
Indeed, the solution to (1) is given by . Since this group consists out of two disjoint components there does not exist a matrix , or continuous nonlinear map for that matter, which can make a vector field akin to (1) globally asymptotically stable. This should be contrasted with the closely related ODE , . Even for the path connected component , The theorem by Sontag obstructs the existence of a continuous global asymptotically stable vector field. This because the group is not simply connected for (This follows most easily from establishing the homotopy equivalence between and via (matrix) Polar Decomposition), hence the fundamental group is nontrivial and contractibility is out of the picture. See that if one would pick to be stable (Hurwitz), then for we have , however .
Following up on Sontag, Bhat and Bernstein revolutionized the field by figuring out some very important ramifications (which are easy to apply). The key observation is the next lemma (that follows from intersection theory arguments, even for nonorientable manifolds).
Lemma [Proposition 1] Compact, boundaryless, manifolds are never contractible.
Clearly, we have to ignore dimensional manifolds here. Important examples of this lemma are the sphere and the rotation group as they make a frequent appearance in mechanical control systems, like a robotic arm. Note that the boundaryless assumption is especially critical here.
Now the main generalization by Bhat and Bernstein with respect to the work of Sontag is to use a (vector) bundle construction of . Loosely speaking, given a base and total manifold , is a vector bundle when for the surjective map and each we have that the fiber is a finitedimensional vector space (see the great book by Abraham, Marsden and Ratiu Section 3.4 and/or the figure below).
Figure 2: A prototypical vector bundle, in this case the cylinder . 
Intuitively, vector bundles can be thought of as manifolds with vector spaces attached at each point, think of a cylinder living in , where the base manifold is and each fiber is a line through extending in the third direction, as exactly what the figure shows. Note, however, this triviality only needs to hold locally.
A concrete example is a rigid body with , for which a large amount of papers claim to have designed continuous globally asymptotically stable controllers. The fact that this was so enormously overlooked makes this topological result not only elegant from a mathematical point of view, but it also shows its practical value. The motivation of this post is precisely to convey this message, topological insights can be very fruitful.
Now we can state their main result
Theorem [Theorem 1] Let be a compact, boundaryless, manifold, being the base of the vector bundle with , then, there is no continuous vector field on with a global asymptotically stable equilibrium point.
Indeed, compactness of can also be relaxed to not being contractible as we saw in the example above, however, compactness is in most settings more tangible to work with.
Example [The Grassmannian manifold] The Grassmannian manifold, denoted by , is the set of all dimensional subspaces . One can identify with the Stiefel manifold , the manifold of all orthogonal frames, in the bundle sense of before, that is such that the fiber represent all frames generating the subspace . In its turn, can be indentified with the compact Lie group (via a quotient), such that indeed is compact. This manifold shows up in several optimization problems and from our continuoustime point of view we see that one can never find, for example, a globally converging gradientflow like algorithm.
Example [Tangent Bundles] By means of a Lagrangian viewpoint, a lot of mechanical systems are of a secondorder nature, this means they are defined on the tangent bundle of some , that is, . However, then, if the configuration manifold is compact, we can again appeal to the Theorem by Bhat and Bernstein. For example, Figure 2 can relate to the manifold over which the pair is defined.
One might wonder, given a vector field over a manifold , since we need contractibility of for to have a global asymptotically stable equilibrium points, how interesting are these manifolds? As it turns out, for , which some might call the highdimensional topological regime, contractible manifolds have a particular structure.
A topological space being simply connected is equivalent to the corresponding fundamental group being trivial. However, to highlight the work by Stallings, we need a slightly less wellknown notion.
Definition [Simply connected at infinity] The topological space is simply connected at infinity, when for any compact subset there exists a where such that is simply connected.
This definition is rather complicated to parse, however, we can give a more tangible description. Let be a noncompact topological space, then, is said to have one end when for any compact there is a where such that is connected. So, , as expected, fails to have one end, while does have one end. Now, Proposition 2.1 shows that if and are simply connected and both have one end, then is simply connected at infinity. This statement somewhat clarifies why dimension and above are somehow easier to parse in the context of topology. A similar statement can be made about the cylindrical space .
Lemma [Product representation Proposition 2.4] Let and be manifolds of dimension greater or equal than . If is contractible and , then, is simply connected at infinity.
Now, most results are usually stated in the context of a piecewise linear (PL) topology. By appealing to celebrated results on triangulization (by Whitehead) we can state the following.
Theorem [Global diffeomorphism Theorem 5.1] Let be a contractible , , dimensional manifold which is simply connected at infinity. If , then, is diffeomorphic to .
Figure 3: The manifold being diffeomorphic to . Then, the benefit of this theorem is that in this setting, dynamical systems can be easily studied without appealing to local charts, describing curve is usually much easier than dealing with directly. 
You might say that this is all obvious, as in Figure 3, we can always do it, also for lowerdimensions. However, there is famous counterexample by Whitehead, which provided lots of motivation for the community:
‘‘There is an open, dimensional manifold which is contractible, but not homeomorphic to .’’
In fact, this example was part of the now notorious program to solve the Poincaré conjecture. All in all, this shows that contractible manifolds are not that interesting from a nonlinear dynamical systems point of view, compact manifolds is a class of manifolds providing for more of a challenge.
So far, the discussion has been on the stabilization of (equilibrium) points, now what if we want to stabilize a curve or any other desirable set, that is, a nontrivial set . It seems like the shape of these objects, with respect to the underlying topology of , must indicate if this is possible again? Let denote an open neighbourhood of some . In what follows, the emphasis is on and having the same topology. We will be brief and follow Moulay and Bhat.
Lemma [Theorem 5] Consider a closedloop dynamical system given rise to a continuous flow. Suppose that the compact set is asymptotically stable with domain of attraction . Then, is a weak deformation retract of .
In rather loose terms, being a weak deformation retract of means that one can continuously ‘‘retract’’ points from and morph them into the set . So, this works for and , due to being punctured. See p.200 for more information on retracts. In Figure 4 below we show that since this lemma is mostly concerned with the local topology, the global topology is less important. For , the set is not a deformation retract of the set such that can never be a region of attraction, this should be contrasted with the setting on the Torus .
Figure 4: The topological relation between and dictates the possibility of being a domain of attraction for . 
The beauty of all of this is that even if you are not so sure about the exact dynamical system at hand, the underlying topology can already provide for some answers. In particular, to determine, what can, or cannot be done.
This post came about during a highly recommended course at EPFL.
Unfortunately the radio silence remained, the good news is that even more cool stuff is on its way*!
For now, we briefly mention a very beautiful interpretation of tensors. Given some vectorspace , let be the set of linear endomorphisms from to . Indeed, for finitedimensional vector spaces one can just think of these maps as being parametrized by matrices. Now, we recall that tensors are multilinear maps which take an element of and its dual space and map it to some real number, that is, .
The claim is that for finitedimensional the set is isomorphic to the set of tensors . This is rather intuitive to see, let and , then, we can write and think of as a matrix. This can be formalized, as is done here by showing there is a bijective map . So, we can identify any with some , that is . Similarly, we can fix and identify with a linear map in the dual space, that is . Regardless, both represent the same tensor evaluation, hence
which is indeed the standard dual map definition.
Now we use this relation, recall that discretetime linear systems are nothing more than maps , that is, linear endomorphims, usually on . Using the tensor interpretation we recover what is sometimes called the ‘‘dual system’’ to , namely (suggestive notation indeed). Using more traditional notation for the primal: and dual: system, we recover that . Then, let and for and being a left and righteigenvector of , respectively. We see that this implies that , for the eigenvalue corresponding to and the eigenvalue corresponding to . Hence, in general, the eigenspaces of and are generically orthogonal.
In other words, we recover the geometric interpretation of mapping volumes under versus .
* See this talk by Daniel.
There was a bit of radio silence, but as indicated here, some interesting stuff is on its way.
In this post we highlight this 1976 paper by Mullis and Roberts on ’Synthesis of Minimum Roundoff Noise Fixed Point Digital Filters’.
Let us be given some singleinput singleoutput (SISO) dynamical system
It is known that the inputoutput behaviour of any , that is, the map from to is invariant under a similarity transform. To be explicit, the tuples and , which correspond to the change of coordinates for some , give rise to the same inputoutput behaviour. Hence, one can define the equivalence relation by imposing that the inputoutput maps of and are equivalent. By doing so we can consider the quotient . However, in practice, given a , is any such that equally useful? For example, the following and are similar, but clearly, is preferred from a numerical point of view:
In what follows, we highlight the approach from Mullis and Roberts and conclude by how to optimally select . Norms will be interpreted in the sense, that is, for , . Also, in what follows we assume that plus that is stable, which will mean and that corresponds to a minimal realization.
The first step is to quantify the error. Assume we have allocated bits at our disposal to present the component of . These values for can differ, but we constrain the average by for some . Let be a 'stepsize’, such that our dynamic range of is bounded by (these are our possible representations). Next we use the modelling choices from Mullis and Roberts, of course, they are debatable, but still, we will infer some nice insights.
First, to bound the dynamic range, consider solely the effect of an input, that is, define by , . Then we will impose the bound on . In combination with the step size (quantization), this results in . Let be a sequence of i.i.d. sampled random variables from . Then we see that . Hence, one can think of as scaling parameter related to the probability with which this dynamic range bound is true.
Next, assume that all the roundoff errors are independent and have a variance of . Hence, the worstcase variance of computating is . To evaluate the effect of this error on the output, assume for the moment that is identically . Then, . Similar to , define as the component of . As before we can compute the variance, this time of the full output signal, which yields . Note, these expressions hinge on the indepedence assumption.
Next, define the (infamous) matrices , by
If we assume that the realization is not just minimal, but that is a controllable pair and that is an observable pair, then, and . Now the key observation is that and similarly . Hence, we can write as and indeed . Using these Lyapunov equations we can say goodbye to the infinitedimensional objects.
To combine these error terms, let say we have apply a coordinate transformation , for some . Specifically, let be diagonal and defined by . Then one can find that , . Where the latter expressions allows to express the output error (variance) by
Now we want to minimize over all plus some optimal selection of . At this point it looks rather painful. To make it work we first reformulate the problem using the wellknown arithmeticgeometric mean inequality for nonnegative sequences :
This inequality yields
See that the right term is independent of , hence this is a lowerbound with respect to minimization over . To achieve this (to make the inequality from an equality), we can select
Indeed, as remarked in the paper, is not per se an integer. Nevertheless, by this selection we find the clean expression from to minimize over systems equivalent to , that is, over some transformation matrix . Define a map by
It turns out that if and only if is diagonal. This follows from Hadamard's inequality. We can use this map to write
Since the term is invariant under a transformation , we can only optimize over a structural change in realization tuple , that is, we need to make and simultaneous diagonal! It turns out that this numerically 'optimal’ realization, denoted , is what is called a principal axis realization.
To compute it, diagonalize , and define . Next, construct the diagonalization . Then our desired transformation is . First, recall that under any the pair becomes . Plugging in our map yields the transformed matrices , which are indeed diagonal.
At last, we do a numerical test in Julia. Consider a linear system defined by the tuple : To simulate numerical problems, we round the maps and to the closest integer where is a noisy sinusoidal input. We compare a perfect (no numerical errors) realization to the rounded, denoted by , naive realization and to the optimized one . We do 100 experiments and show the mean plus the full confidence interval (of the inputoutput behaviour), the optimized representation is remarkably better. Numerically we observe that for we have , which is precisely where the naive struggles. 
: To show this AMGM inequality, we can use Jensen's inequality for concave functions, that is, . We can use the logarithm as our prototypical concave function on and find . Then, the result follows.
: The inequality attributed to Hadamard is slightly more difficult to show. In its general form the statement is that , for the column of . The inequality becomes an equality when the columns are mutually orthogonal. The intuition is clear if one interprets the determinant as the signed volume spanned by the columns of , which. In case , we know that there is such that , hence, by this simple observation it follows that
Equality holds when the columns are orthogonal, so must be diagonal, but must also hold, hence, must be diagonal, and thus must be diagonal, which is the result we use.
As a follow up to the previous post, we discuss a lower bound, given as exercise P7.1.14 in (GvL13). I have never seen this bound before, especially not with the corresponding proof, so here we go. Given any , we have the following lower bound:
which can be thought of as a sharpening of the bound . In the case of a singular , the bound is trivial , but in the general setting it is less obvious. To prove that this holds we need a few ingredients. First, we need that for any we have . Then, we construct two decompositions of , (1) the Jordan Normal form , and (2) the Singular Value Decomposition . Using the multiplicative property of the determinant, we find that
Hence, since it follows that and thus we have the desired result.
(Update 8 March) It is of course interesting to get insight in how sharp this bound is, in general. To that end we consider random matrices with . When we compute the relative error for samples we obtain the histogram as shown on the left. Clearly, the relative error is not concentrated near . Nevertheless, it is an interesting bound, from a theoretical point of view. 
(GvL13) G.H. Golub and C.F. Van Loan: ‘‘Matrix Computations’’, 2013 John Hopkins University Press.
Lately, linear discretetime control theory has start to appear in areas far beyond the ordinary. As a byproduct, papers start to surface which claim that stability of linear discretetime systems
is characterized by . The confunsion is complete by calling this object the spectral norm of a matrix . Indeed, for fixed coordinates, stability of is not characterized by , but by . If , then, all absolute eigenvalues of are strictly smaller than one, and hence for , . This follows from the Jordan form of in combination with the simple observation that for , .
To continue, define two sets; and . Since we have . Now, the main question is, how much do we lose by approximating with ? Motivation to do so is given by the fact that is a norm and hence a convex function, i.e., when given a convex polytope with vertices , if , then . Note that is not a norm, can be without (consider an uppertriangular matrix with zeros on the main diagonal), the triangle inequality can easily fail as well. For example, let
Then , but , hence fails to hold. A setting where the aforementioned property of might help is Robust Control, say we want to assess if some algorithm rendered a compact convex set , filled with 's, stable. As highlighted before, we could just check if all extreme points of are members of , which might be a small and finite set. Thus, computationally, it appears to be attractive to consider over the generic .
As a form of critique, one might say that is a lot larger than . Towards such an argument, one might recall that . Indeed, if , but . Therefore, it seems like the set of for which considering over is reasonable, is negligibly small. To say a bit more, since we see that we can always find a ball with nonzero volume fully contained in . Hence, is at least locally dense in . So in principle we could try to investigate . For , the sets agree, which degrades asymptotically. However, is this the way to go? Lets say we consider the set . Clearly, the sets and are different, even in volume, but for sufficiently large , should we care? The behaviour they parametrize is effectively the same.
We will stress that by approximating with , regardless of their volumetric difference, we are ignoring a full class of systems and miss out on a nonneglible set of behaviours. To see this, any system described by is contractive in the sense that , while systems in are merely asymptotically stable. They might wander of before they return, i.e., there is no reason why for all we must have . We can do a quick example, consider the matrices
Then , and both and have . We observe that indeed is contractive, for any initial condition on , we move strictly inside the sphere, whereas for , when starting from the same initial condition, we take a detour outside of the sphere before converging to . So although and have the same spectrum, they parametrize different systems.
In our deterministic setting this would mean that we would confine our statespace to a (solid) sphere with radius , instead of . Moreover, in linear optimal control, the resulting closedloop system is usually not contractive. Think of the infamous pendulum on a cart. Being energy efficient has usually nothing to do with taking the shortest, in the Euclidean sense, path.
(update June 06): As suggested by Pedro Zattoni, we would like to add some nuance here. As highlighted in the first 12 minutes of this lecture by Pablo Parrilo, if is stable, then, one can always find a (linear) change of coordinates such that the transformed system matrix is contractive. Although a very neat observation, you have to be careful, since your measurements might not come from this transformed system. So either you assume merely that is stable, or you assume that is contractive, but then you might need to find an additional map, mapping your states to your measurements.
: Recall, . Then, let be some eigenvector of . Now we have . Since this eigenvalue is arbitrary it follows that .
: Let then . This follows from the being a norm.
: Clearly, if , we have . Now, when , does this imply that ? The answer is no, consider
Then, , yet, . For the full set of conditions on such that see this paper by Goldberg and Zwas.
: Recall that . This expression is clearly larger or equal to .
One of the most famous problems in linear control theory is that of designing a control law which minimizes some cost being quadratic in input and state. We all know this as the Linear Quadratic Regulator (LQR) problem. There is however one problem (some would call it a blessing) with this formulation once the dynamics contain some zeromean noise: the control law is independent of this stochasticity. This is easy in proofs, easy in practice, but does it work well? Say, your control law is rather slow, but bringing you back to in the end of time. In the noiseless case, no problem. But now say there is a substantial amount of noise, do you still want such a slow control law? The answer is most likely no since you quickly drift away. The classical LQR formulation does not differentiate between noise intensities and hence can be rightfully called naive as Bertsekas did (Bert76). Unfortunately, this name did not stuck.
Can we do better? Yes. Define the cost function
and consider the problem of finding the minimizing policy in:
Which is precisely the classical LQR problem, but now with the cost wrapped in an exponential utility function parametrized by . This problem was pioneered by most notably Peter Whittle (Wh90).
Before we consider solving this problem, let us interpret the risk parameter . For we speak of a risksensitive formulation while for we are riskseeking. This becomes especially clear when you solve the problem, but a quick way to see this is to consider the approximation of near , which yields , so relates to pessimism and to optimism indeed. Here we skipped a few scaling factors, but the idea remains the same, for a derivation, consider cumulant generating functions and have a look at our problem.
As with standard LQR, we would like to obtain a stabilizing policy and to that end we will be mostly bothered with solving . However, instead of immediately trying to solve the infinite horizon average cost Bellman equation it is easier to consider for finite first. Then, when we can prove monotonicty and upperbound , the infinite horizon optimal policy is given by . The reason being that monotonic sequences which are uniformly bounded converge.
The main technical tool towards finding the optimal policy is the following Lemma similar to one in the Appendix of (Jac73):
Lemma
Consider a noisy linear dynamical system defined by with and let be shorthand notation for . Then, if holds we have
where .
Proof
Let and recall that the shorthand notation for is , then:
Here, the first step follows directly from being a zeromean Gaussian. In the second step we plug in . Then, in the third step we introduce a variable with the goal of making the covariance matrix of a Gaussian with mean . We can make this work for
and additionally
Using this approach we can integrate the latter part to and end up with the final expression. Note that in this case the random variable needs to be Gaussian, since the second to last expression in equals by being a Gaussian probability distribution integrated over its entire domain.
What is the point of doing this? Let and assume that represents the costtogo from stage and state . Then consider
Note, since we work with a sum within the exponent, we must multiply within the righthandside of the Bellman equation. From there it follows that , for
The key trick in simplifying your expressions is to apply the logarithm after minimizing over such that the fraction of determinants becomes a stateindependent affine term in the cost. Now, using a matrix inversion lemma and the pushthrough rule we can remove and construct a map :
such that . See below for the derivations, many (if not all) texts skip them, but if you have never applied the pushthrough rule they are not that obvious.
As was pointed out for the first time by Jacobson (Jac73), these equations are precisely the ones we see in (noncooperative) Dynamic Game theory for isotropic and appropriately scaled .
Especially with this observation in mind there are many texts which show that is welldefined and finite, which relates to finite cost and a stabilizing control law . To formalize this, one needs to assume that is a minimal realization for defined by . Then you can appeal to texts like (BB95).
Numerical Experiment and Robustness Interpretation
To show what happens, we do a small dimensional example. Here we want to solve the RiskSensitive ) and the Riskneutral () infinitehorizon average cost problem for
, , . There is clearly a lot of noise, especially on the second signal, which also happens to be critical for controlling the first state. This makes it interesting. We compute and . Given the noise statistics, it would be reasonable to not take the certainty equivalence control law since you control the first state (which has little noise on its line) via the second state (which has a lot of noise on its line). Let be the state under and the state under .
We see in the plot below (for some arbitrary initial condition) typical behaviour, does take the noise into account and indeed we see the induces a smaller variance.
So, is more robust than in a particular way. It turns out that this can be neatly explained. To do so, we have to introduce the notion of Relative Entropy (KullbackLeibler divergence). We will skip a few technical details (see the references for full details). Given a measure on , then for any other measure , being absolutely continuous with respect to (), define the Relative Entropy as:
Now, for any measurable function on , being bounded from below, it can be shown (see (DP96)) that
For the moment, think of as your standard finitehorizon LQR cost with product measure , then we see that an exponential utility results in the understanding that a control law which minimizes is robust against adversarial noise generated by distributions sufficiently close (measured by ) to the reference .
Here we skipped over a lot of technical details, but the intuition is beautiful, just changing the utility to the exponential function gives a wealth of deep distributional results we just touched upon in this post.
Simplification steps in
We only show to get the simpler representation for the input, the approach to obtain is very similar.
First, using the matrix inversion lemma:
we rewrite into:
Note, we factored our since we cannot assume that is invertible. Our next tool is called the pushthrough rule. Given and we have
You can check that indeed . Now, to continue, plug this expression for into the input expression:
Indeed, we only used factorizations and the pushthrough rule to arrive here.
(Bert76) Dimitri P. Bertsekas : ‘‘Dynamic Programming and Stochastic Control’’, 1976 Academic Press.
(Jac73) D. Jacobson : ‘‘Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games’’, 1973 IEEE TAC.
(Wh90) Peter Whittle : ‘‘Risksensitive Optimal Control’’, 1990 Wiley.
(BB95) Tamer Basar and Pierre Bernhard : ‘‘Optimal Control and Related Minimax Design Problems A Dynamic Game Approach’’ 1995 Birkhauser.
(DP96) Paolo Dai Pra, Lorenzo Meneghini and Wolfgang J. Runggaldier :‘‘Connections Between Stochastic Control and Dynamic Games’’ 1996 Math. Control Signals Systems.
Theo Jansen, designer of the infamous ‘‘strandbeesten’’ explains in a relatively recent series of videos that the design of his linkage system is the result of what could be called an early implementation of genetic programming. Specifically, he wanted the profile that the foot follows to be flat on the bottom. He does not elaborate on this too much, but the idea is that in this way his creatures flow over the surface instead of the more bumpy motion we make. Moreover, you can imagine that in this way the center of mass remains more or less at a constant height, which is indeed energy efficient. Now, many papers have been written on understanding his structure, but I find most of them rather vague, so lets try a simple example.
To that end, consider just 1 leg, 1 linkage system as in the figure(s) below. We can write down an explicit expression for where and . See below for an explicit derivation. Now, with his motivation in mind, what would be an easy and relevant cost function? An idea is to maximize the ratio , of course, subject to still having a functioning leg. To that end we consider the cost function (see the schematic below for notation):
Of course, this is a heuristic, but the intuition is that this should approximate . Let be the set of parameters as given in Jansen his video, then we apply a few steps of (warmstart) gradient ascent: , .
In the figure on the left we compare the linkage as given by Jansen (in black) with the result of just two steps of gradient ascent (in red). You clearly see that the trajectory of the foot becomes more shallow. Nevertheless, we see that the profile starts to become curved on the bottom. It is actually interesting to note that , which is still far from but already remarkably small for . 
Can we conclude anything from here? I would say that we need dynamics, not just kinematics, but more importantly, we need to put the application in the optimization and only Theo Jansen understands the constraints of the beach.
Still, it is fun to see for yourself how this works and like I did you can build a (baby) strandbeest yourself using nothing more than some PVC tubing and a source of heat.
Derivation of :
Essentially, all that you need is the cosinerule plus some feeling for how to make the equations continuous.
To start, let and (just the standard unit vectors), plus let be a standard (counterclockwise) rotation matrix acting on . To explain the main approach, when we want to find the coordinates of for example point , we try to compute the angle such that when we rotate the unit vector with length emanating from point , we get point . See the schematic picture below.
Then we easily obtain , and . Now, to obtain and compute the length of the diagonal, and using the cosine rule , . Then, the trick is to use to compute . To avoid angles larger than , consider an inner product with and add the remaining ; . From there we get , , , . The point is not missing, to check your code it is convenient to compute and check that agrees with for all . In a similar fashion, , , . Then, for , again compute a diagonal plus , , such that for we have . At last, , such that for .
This short note is inspired by the beautiful visualization techniques from Duistermaat and Kolk (DK1999) (themselves apparently inspired by Atiyah).
Let's say we have a dimensional linear discretetime dynamical system , which preserves the volume of the cube under the dynamics, i.e. for any .
Formally put, this means that is part of a certain group, specifically, consider some field and define the Special Linear group by
Now, assume that we are only interested in matrices such that the cube remains bounded under the dynamics, i.e., is bounded. In this scenario we restrict our attention to . To see why this is needed, consider and both with determinant :
If we look at the image of under both these maps (for several iterations), we see that volume is preserved, but also clearly that the set is extending indefinitely in the horizontal direction.
To have a somewhat interesting problem, assume we are given with while it is our task to find a such that , hence, find feedback which not only preserves the volume, but keeps any set of initial conditions (say ) bounded over time.
Towards a solution, we might ask, given any what is the nearest (in some norm) rotation matrix? This turns out to be a classic question, starting from orthogonal matrices, the solution to is given by , where follows from the SVD of , . Differently put, when we use a polar decomposition of , , then the solution is given by . See this note for a quick multiplier proof. An interesting sidenote, problem is welldefined since is compact. To see this, recall that for any we have , hence the set is bounded, plus is the preimage of under , , hence the set is closed as well.
This is great, however, we would like to optimize over instead. To do so, one usually resorts to simply checking the sign  and if necessary  flipping it via finding the closest matrix with a change in determinant sign. We will see that by selecting appropriate coordinates we can find the solution without this checking of signs.
For today we look at and largely follow (DK1999), for such a dimensional matrix, the determinant requirement translates to . Under the following (invertible) linear change of coordinates
becomes , i.e., for any pair the point runs over a circle with radius . Hence, we obtain the diffeomorphism . We can use however a more pretty diffeomorphism of the sphere and an open set in . To that end, use the diffeomorphism:
for (formal way of declaring that should not be any real number) and the pair being part of (the open unitdisk). To see this, recall that the determinant is not a linear operator. Since we have a dimensional example we can explicitly compute the eigenvalues of any (plug into the characteristic equation) and obtain:
At this point, the only demand on the eigenvalues is that . Therefore, when we would consider within the context of a discrete linear dynamical system , is either a saddlepoint, or we speak of a marginally stable system (). We can visualize the set of all marginally stable , called , defined by all satisfying
(DK1999) J.J. Duistermaat and J.A.C. Kolk : ‘‘Lie Groups’’, 1999 Springer.
Previously we looked at ,  and its resulting flow  as the result from mapping to the sphere. However, we saw that for this flow convergences to , with such that for we have . Hence, it is interesting to look at the flow from an optimization point of view: .
A fair question would be, is our flow not simply Riemannian gradient ascent for this problem? As common with these kind of problems, such a hypothesis is something you just feel.
Now, using the tools from (CU1994, p.311) we can compute the gradient of (on the sphere) via , where is a vector field normal to , e.g., . From there we obtain
To make our life a bit easier, use instead of the map . Moreover, set . Then it follows that
Of course, to make the computation cleaner, we changed to , but the relation between and is beautiful. Somehow, mapping trajectories of , for some , to the sphere corresponds to (Riemannian) gradient ascent applied to the problem .
To do an example, let be given by We can compare with . We see that the gradient flow under the quadratic function takes a clearly ‘‘shorter’’ path. 
Now, we formalize the previous analysis a bit a show how fast we convergence. Assume that the eigenvectors are ordered such that eigenvector corresponds to the largest eigenvalue of . Then, the solution to is given by
Let be expressed in eigenvector coordinates, with all (normalized). Moreover, assume all eigenvalues are distinct. Then, to measure if is near , we compute , which is if and only if is parallel to . To simplify the analysis a bit, we look at , for some perturbation , this yields
Next, take the the (natural) logarithm on both sides:
This logsumexp terms are hard to deal with, but we can apply the socalled ‘‘logsumexp trick’’:
In our case, we set and obtain
We clearly observe that for the LHS approaches from below, which means that from above, like intended. Of course, we also observe that the mentioned method is not completely general, we already assume distinct eigenvalues, but there is more. We do also not convergence when , which is however a set of measure on the sphere .
More interestingly, we see that the convergence rate is largely dictated by the ‘‘spectral gap/eigengap’’ . Specifically, to have a particular projection error , such that , we need
Comparing this to the resulting flow from , , we see that we have the same flow, but with . This is interesting, since and have the same eigenvectors, yet a different (scaled) spectrum. With respect to the convergence rate, we have to compare and for any with (the additional is not so interesting).
It is obvious what we will happen, the crux is, is larger or smaller than ? Can we immediately extend this to a Newtontype algorithm? Well, this fails (globally) since we work in instead of purely with . To be concrete, , we never have degrees of freedom.
Of course, these observations turned out to be far from new, see for example (AMS2008, sec. 4.6).
(AMS2008) P.A. Absil, R. Mahony and R. Sepulchre: ‘‘Optimization Algorithms on Matrix Manifolds’’, 2008 Princeton University Press.
(CU1994) Constantin Udriste: ‘‘Convex Functions and Optimization Methods on Riemannian Manifolds’’, 1994 Kluwer Academic Publishers.
Using the ideas from the previous post, we immediately obtain a dynamical system to compute for any (symmetric positive definite matrices). Here we use the vector field , defined by and denote a solution by , where is the initial condition. Moreover, define a map by . Then , where is the eigenvector corresponding to the largest eigenvalue.
We can do a simple example, let with . Indeed, when given 100 , we observe convergence. To apply this in practice we obviously need to discretize the flow. To add to that, if we consider the discretetime version of , we can get something of the form , which is indeed the Power Iteration algorithm, used to compute the largest eigenvalue of . Recall however that this algorithm needs stronger conditions (on ) to work than our flow. 
In this post we start a bit with control of systems living on compact metric spaces. This is interesting since we have to rethink what we want, cannot even occur. Let a great arc (circle) on be parametrized by the normal of a hyperplane through , i.e., , is the axis of rotation for . 
To start, consider a linear dynamical systems , , for which the solution is given by . We can map this solution to the sphere via . The corresponding vector field for is given by
The beauty is the following, to find the equilibria, consider . Of course, is not valid since . However, take any eigenvector of , then is still an eigenvector, plus it lives on . If we plug such a scaled (from now on consider all eigenvectors to live on ) into we get , which is rather cool. The eigenvectors of are (at least) the equilibria of . Note, if , then so is , each eigenvector gives rise to two equilibria. We say at least, since if is nontrivial, then for any , we get .
Let , then what can be said about the stability of the equilibrium points? Here, we must look at , where the dimensional system is locally a dimensional linear system. To do this, we start by computing the standard Jacobian of , which is given by
As said before, we do not have a dimensional system. To that end, we assume with diagonalization (otherwise peform a GramSchmidt procedure) and perform a change of basis . Then if we are interested in the qualitative stability properties of (the eigenvector of ), we can simply look at the eigenvalues of the matrix resulting from removing the row and column in , denoted .
We do two examples. In both figures are the initial conditions indicated with a dot while the equilibrium points are stars.
First, let Clearly, for we have and let be . Then using the procedure from before, we find that is stable (attracting), while are locally unstable in one direction while in the other (along the great arc). Hence, this example gives rise to the two stable poles. 
Consider an unknown dynamical system parametrized by . In this short note we look at a relatively simple, yet neat method to assess stability of via finite calls to a particular oracle. Here, we will follow the notes by Roman Vershynin (RV2011).
Assume we have no knowledge of , but we are able to obtain for any desired . It turns out that we can use this oracle to bound from above.
First, recall that since is symmetric, . Now, the common definition of is . The constraint set is not per se convenient from a computational point of view. Instead, we can look at a compact constraint set: , where is the sphere in . The latter formulation has an advantage, we can discretize these compact constraints sets and tame functions over the individual chunks.
To do this formally, we can use nets, which are convenient in quantitative analysis on compact metric spaces . Let be a net for if . Hence, we measure the amount of balls to cover .
For example, take as our metric space where we want to bound (the cardinality, i.e., the amount of balls). Then, to construct a minimal set of balls, consider an net of balls , with . Now the insight is simply that . Luckily, we understand the volume of dimensional Euclidean balls well such that we can establish
Which is indeed sharper than the bound from Lemma 5.2 in (RV2011).
Next, we recite Lemma 5.4 from (RV2011). Given an net for with we have
This can be shown using the CauchySchwartz inequality. Now, it would be interesting to see how sharp this bound is. To that end, we do a dimensional example (). Note, here we can easily find an optimal grid using polar coordinates. Let the unknown system be parametrized by
such that . In the figures below we compute the upperbound to for several nets . Indeed, for finer nets, we see the upperbound approaching . However, we see that the convergence is slow while being inversely related to the exponential growth in .
Overall, we see that after about a 100 calls to our oracle we can be sure about being exponentially stable. Here we just highlighted an example from (RV2011), at a later moment we will discuss its use in probability theory.
In this first post we will see that the ‘‘almost surely’’ in the title is not only hinting at stochasticity.
For a long time this picture  as taken from Gabriel Peyré his outstanding twitter feed  has been on a wall in our local wildlife (inc. hedgehogs) shelter. This has lead to a few fun discussions and here we try to explain the picture a bit more.
Let . Now, we want to find the largest ball within . We can find that is given by with . This follows the most easily from geometric intuition.
To prove it, see that has ‘‘similar’’ faces, without loss of generality we take the one in the positive orthant of . Now we fit a dimensional plane (hyperplane) to this face and find the point the closest (in norm) to . This hyperplane can be parametrized by for and . Thus the normal (direction) is given by . To find , observe that for we have . Hence we have . Now, let and define the other unit vectors correspondingly. Then, since for any we can think of a (skinny) hedgehog indeed for . The inscribed ball shrinks to , while the spikes remain. 
Besides being a fun visualization of concentration, this picture means a bit more to me. For years I have been working at the Wildlife shelter in Delft and the moment has come where the lack of financial support from surrounding municipalities is becoming critical. As usual, this problem can be largely attributed to bureaucracy and the exorbitant amount of managers this world has. Nevertheless, the team in Delft and their colleagues around the country do a fantastic job in spreading the word (see Volkskrant, RTLnieuws). Lets hope Delft, and it neighbours, finally see value in protecting the local wildlife we still have.