Section: New Results

Stability and uncertain dynamics: new results

Let us list here our new results about stability and stabilization of control systems, on the properties of systems with uncertain dynamics.

• In [8] we consider a one-dimensional controlled reaction-diffusion equation, where the control acts on the boundary and is subject to a constant delay. Such a model is a paradigm for more general parabolic systems coupled with a transport equation. We prove that it is possible to stabilize (in $H^1$ norm) this process by means of an explicit predictor-based feedback control that is designed from a finite-dimensional subsystem. The implementation is very simple and efficient and is based on standard tools of pole-shifting. Our feedback acts on the system as a finite-dimensional predictor. We compare our approach with the backstepping method.

• In [14] we consider the one dimensional Schrödinger equation with a bilinear control and prove the rapid stabilization of the linearized equation around the ground state. The feedback law ensuring the rapid stabilization is obtained using a transformation mapping the solution of the linearized equation to the solution of an exponentially stable target linear equation. A suitable condition is imposed on the transformation in order to cancel the non-local terms arising in the kernel system. This conditions also insures the uniqueness of the transformation. The continuity and invertibility of the transformation follows from exact controllability of the linearized system.

• Based on the notion of generalized homogeneity, we develop in [17] a new algorithm of feedback control design for a plant modeled by a linear evolution equation in a Hilbert space with a possibly unbounded operator. The designed control law steers any solution of the closed-loop system to zero in a finite time. Method of homogeneous extension is presented in order to make the developed control design principles to be applicable for evolution systems with non-homogeneous operators. The design scheme is demonstrated for heat equation with the control input distributed on the segment $[0,1]$.

• In [19] we analyse the asymptotic behaviour of integro-differential equations modeling $N$ populations in interaction, all structured by different traits. Interactions are modeled by non-local terms involving linear combinations of the total number of individuals in each population. These models have already been shown to be suitable for the modeling of drug resistance in cancer, and they generalise the usual Lotka–Volterra ordinary differential equations. Our aim is to give conditions under which there is persistence of all species. Through the analysis of a Lyapunov function, our first main result gives a simple and general condition on the matrix of interactions, together with a convergence rate. The second main result establishes another type of condition in the specific case of mutualistic interactions. When either of these conditions is met, we describe which traits are asymptotically selected.

• The goal of [20] is to compute a boundary control of reaction-diffusion partial differential equation. The boundary control is subject to a constant delay, whereas the equation may be unstable without any control. For this system equivalent to a parabolic equation coupled with a transport equation, a prediction-based control is explicitly computed. To do that we decompose the infinite-dimensional system into two parts: one finite-dimensional unstable part, and one stable infinite-dimensional part. A finite-dimensional delay controller is computed for the unstable part, and it is shown that this controller succeeds in stabilizing the whole partial differential equation. The proof is based on a an explicit form of the classical Artstein transformation, and an appropriate Lyapunov function. A numerical simulation illustrate the constructive design method.

• [27] focuses on the (local) small-time stabilization of a Korteweg-de Vries equation on bounded interval, thanks to a time-varying Dirichlet feedback law on the left boundary. Recently, backstepping approach has been successfully used to prove the null controllability of the corresponding linearized system, instead of Carleman inequalities. We use the “adding an integrator" technique to gain regularity on boundary control term which clears the difficulty from getting stabilization in small-time.

• Motivated by improved ways to disrupt brain oscillations linked to Parkinson's disease, we propose in [29] an adaptive output feedback strategy for the stabilization of nonlinear time-delay systems evolving on a bounded set. To that aim, using the formalism of input-to-output stability (IOS), we first show that, for such systems, internal stability guarantees robustness to exogenous disturbances. We then use this feature to establish a general result on scalar adaptive output feedback of time-delay systems inspired by the “$\sigma$-modification" strategy. We finally apply this result to a delayed neuronal population model and assess numerically the performance of the adaptive stimulation.

• In [35] we consider open channels represented by Saint-Venant equations that are monitored and controlled at the downstream boundary and subject to unmeasured flow disturbances at the upstream boundary. We address the issue of feedback stabilization and disturbance rejection under Proportional-Integral (PI) boundary control. For channels with uniform steady states, the analysis has been carried out previously in the literature with spectral methods as well as with Lyapunov functions in Riemann coordinates. In [35], our main contribution is to show how the analysis can be extended to channels with non-uniform steady states with a Lyapunov function in physical coordinates.

• In [37], we study the exponential stabilization of a shock steady state for the inviscid Burgers equation on a bounded interval. Our analysis relies on the construction of an explicit strict control Lyapunov function. We prove that by appropriately choosing the feedback boundary conditions, we can stabilize the state as well as the shock location to the desired steady state in $H^2$-norm, with an arbitrary decay rate.

• Given a discrete-time linear switched system $\Sigma \left(A\right)$ associated with a finite set $A$ of matrices, we consider in [40] the measures of its asymptotic behavior given by, on the one hand, its deterministic joint spectral radius ${\rho }_d\left(A\right)$ and, on the other hand, its probabilistic joint spectral radii ${\rho }_p(v,P,A)$ for Markov random switching signals with transition matrix $P$ and a corresponding invariant probability $v$. Note that ${\rho }_d\left(A\right)$ is larger than or equal to ${\rho }_p(v,P,A)$ for every pair $(v,P)$. In this paper, we investigate the cases of equality of ${\rho }_d\left(A\right)$ with either a single ${\rho }_p(v,P,A)$ or with the supremum of ${\rho }_p(v,P,A)$ over $(v,P)$ and we aim at characterizing the sets $A$ for which such equalities may occur.

• In [41], we introduce a method to get necessary and sufficient stability conditions for systems governed by 1-D nonlinear hyperbolic partial-differential equations with closed-loop integral controllers, when the linear frequency analysis cannot be used anymore. We study the stability of a general nonlinear transport equation where the control input and the measured output are both located on the boundaries. The principle of the method is to extract the limiting part of the stability from the solution using a projector on a finite-dimensional space and then use a Lyapunov approach. We improve a result of Trinh, Andrieu and Xu, and give an optimal condition for the design of the controller. The results are illustrated with numerical simulations where the predicted stable and unstable regions can be clearly identified.

• In [44] we construct explicit time-varying feedback laws leading to the global (null) stabilization in small time of the viscous Burgers equation with three scalar controls. Our feedback laws use first the quadratic transport term to achieve the small-time global approximate stabilization and then the linear viscous term to get the small-time local stabilization.

• In [46] we address the question of the exponential stability for the $C^1$ norm of general 1-D quasilinear systems with source terms under boundary conditions. To reach this aim, we introduce the notion of basic $C^1$ Lyapunov functions, a generic kind of exponentially decreasing function whose existence ensures the exponential stability of the system for the $C^1$ norm. We show that the existence of a basic $C^1$ Lyapunov function is subject to two conditions: an interior condition, intrinsic to the system, and a condition on the boundary controls. We give explicit sufficient interior and boundary conditions such that the system is exponentially stable for the $C^1$ norm and we show that the interior condition is also necessary to the existence of a basic $C^1$ Lyapunov function. Finally, we show that the results conducted in this article are also true under the same conditions for the exponential stability in the $C^p$ norm, for any $p\ge 1$.

• In [47] we study the exponential stability for the $C^1$ norm of general 2$\times$2 1-D quasilinear hyperbolic systems with source terms and boundary controls. When the eigenvalues of the system have the same sign, any nonuniform steady-state can be stabilized using boundary feedbacks that only depend on measurements at the boundaries and we give explicit conditions on the gain of the feedback. In other cases, we exhibit a simple numerical criterion for the existence of basic $C^1$ Lyapunov function, a natural candidate for a Lyapunov function to ensure exponential stability for the $C^1$ norm. We show that, under a simple condition on the source term, the existence of a basic $C^1$ (or $C^p$ , for any $p\ge 1$) Lyapunov function is equivalent to the existence of a basic $H^2$ (or $H^q$ , for any $q\ge 2$) Lyapunov function, its analogue for the $H^2$ norm. Finally, we apply these results to the nonlinear Saint-Venant equations. We show in particular that in the subcritical regime, when the slope is larger than the friction, the system can always be stabilized in the $C^1$ norm using static boundary feedbacks depending only on measurements of at the boundaries, which has a large practical interest in hydraulic and engineering applications.

• In [48] we study the exponential stability in the $H^2$ norm of the nonlinear Saint-Venant (or shallow water) equations with arbitrary friction and slope using a single Proportional-Integral (PI) control at one end of the channel. Using a local dissipative entropy we find a simple and explicit condition on the gain the PI control to ensure the exponential stability of any steady-states. This condition is independent of the slope, the friction, the length of the river, the inflow disturbance and, more surprisingly, the steady-state considered. When the inflow disturbance is time-dependent and no steady-state exist, we still have the Input-to-State stability of the system, and we show that changing slightly the PI control enables to recover the exponential stability of slowly varying trajectories.

• The exponential stability problem of the nonlinear Saint-Venant equations is addressed in [49]. We consider the general case where an arbitrary friction and space-varying slope are both included in the system, which lead to non-uniform steady-states. An explicit quadratic Lyapunov function as a weighted function of a small perturbation of the steady-states is constructed. Then we show that by a suitable choice of boundary feedback controls, that we give explicitly, the local exponential stability of the nonlinear Saint-Venant equations for the $H^2$-norm is guaranteed.

• [53] elaborates control strategies to prevent clustering effects in opinion formation models. This is the exact opposite of numerous situations encountered in the literature where, on the contrary, one seeks controls promoting consensus. In order to promote declustering, instead of using the classical variance that does not capture well the phenomenon of dispersion, we introduce an entropy-type functional that is adapted to measuring pairwise distances between agents. We then focus on a Hegselmann-Krause-type system and design declustering sparse controls both in finite-dimensional and kinetic models. We provide general conditions characterizing whether clustering can be avoided as function of the initial data. Such results include the description of black holes (where complete collapse to consensus is not avoidable), safety zones (where the control can keep the system far from clustering), basins of attraction (attractive zones around the clustering set) and collapse prevention (when convergence to the clustering set can be avoided).

• In [54] we consider the problem of controlling parabolic semilinear equations arising in population dynamics, either in finite time or infinite time. These are the monostable and bistable equations on $(0,L)$ for a density of individuals $0\le y(t,x)\le 1$, with Dirichlet controls taking their values in $[0,1]$. We prove that the system can never be steered to extinction (steady state 0) or invasion (steady state 1) in finite time, but is asymptotically controllable to 1 independently of the size $L$, and to 0 if the length $L$ of the interval domain is less than some threshold value $L^{*}$, which can be computed from transcendental integrals. In the bistable case, controlling to the other homogeneous steady state $0<\theta <1$ is much more intricate. We rely on a staircase control strategy to prove that $\theta$ can be reached in finite time if and only if $L<L^{?}$. The phase plane analysis of those equations is instrumental in the whole process. It allows us to read obstacles to controllability, compute the threshold value for domain size as well as design the path of steady states for the control strategy.

• Given a linear control system in a Hilbert space with a bounded control operator, we establish in [56] a characterization of exponential stabilizability in terms of an observability inequality. Such dual characterizations are well known for exact (null) controllability. Our approach exploits classical Fenchel duality arguments and, in turn, leads to characterizations in terms of observability inequalities of approximately null controllability and of $\alpha$-null controllability. We comment on the relationships between those various concepts, at the light of the observability inequalities that characterize them.

• In [58] we use the backstepping method to study the stabilization of a 1-D linear transport equation on the interval $(0,L)$, by controlling the scalar amplitude of a piecewise regular function of the space variable in the source term. We prove that if the system is controllable in a periodic Sobolev space of order greater than 1, then the system can be stabilized exponentially in that space and, for any given decay rate, we give an explicit feedback law that achieves that decay rate.

Let us also mention the lecture notes [31] on stabilization of semilinear PDE's, which have been published this year.