No Arabic abstract
A nonlinear control system is said to be weakly contractive in the control if the flow that it generates is non-expanding (in the sense that the distance between two trajectories is a non-increasing function of time) for some fixed Riemannian metric independent of the control. We prove in this paper that for such systems, local asymptotic stabilizability implies global asymptotic stabilizability by means of a dynamic state feedback. We link this result and the so-called Jurdjevic and Quinn approach.
The aim of this paper is to prove the superexponential stabilizability to the ground state solution of a degenerate parabolic equation of the form begin{equation*} u_t(t,x)+(x^{alpha}u_x(t,x))_x+p(t)x^{2-alpha}u(t,x)=0,qquad tgeq0,xin(0,1) end{equation*} via bilinear control $pin L_{loc}^2(0,+infty)$. More precisely, we provide a control function $p$ that steers the solution of the equation, $u$, to the ground state solution in small time with doubly-exponential rate of convergence. The parameter $alpha$ describes the degeneracy magnitude. In particular, for $alphain[0,1)$ the problem is called weakly degenerate, while for $alphain[1,2)$ strong degeneracy occurs. We are able to prove the aforementioned stabilization property for $alphain [0,3/2)$. The proof relies on the application of an abstract result on rapid stabilizability of parabolic evolution equations by the action of bilinear control. A crucial role is also played by Bessels functions.
We provide out-of-sample certificates on the controlled invariance property of a given set with respect to a class of black-box linear systems. Specifically, we consider linear time-invariant models whose state space matrices are known only to belong to a certain family due to a possibly inexact quantification of some parameters. By exploiting a set of realizations of those undetermined parameters, verifying the controlled invariance property of the given set amounts to a linear program, whose feasibility allows us to establish an a-posteriori probabilistic certificate on the controlled invariance property of such a set with respect to the nominal linear time-invariant dynamics. The proposed framework is applied to the control of a networked multi-agent system with unknown weighted graph.
We consider a general multi-armed bandit problem with correlated (and simple contextual and restless) elements, as a relaxed control problem. By introducing an entropy premium, we obtain a smooth asymptotic approximation to the value function. This yields a novel semi-index approximation of the optimal decision process, obtained numerically by solving a fixed point problem, which can be interpreted as explicitly balancing an exploration-exploitation trade-off. Performance of the resulting Asymptotic Randomised Control (ARC) algorithm compares favourably with other approaches to correlated multi-armed bandits.
The long-time average behaviour of the value function in the calculus of variations, where both the Lagrangian and Hamiltonian are Tonelli, is known to be connected to the existence of the limit of the corresponding Abel means as the discount factor goes to zero. Still in the Tonelli case, such a limit is in turn related to the existence of solutions of the critical (or, ergodic) Hamilton-Jacobi equation. The goal of this paper is to address similar issues when the Hamiltonian fails to be Tonelli: in particular, for control systems that can be associated with a family of vector fields which satisfies the Lie Algebra rank condition. First, following a dynamical approach we characterise the unique constant for which the ergodic equation admits solutions. Then, we construct a critical solution which coincides with its Lax-Oleinik evolution.
We prove rapid stabilizability to the ground state solution for a class of abstract parabolic equations of the form begin{equation*} u(t)+Au(t)+p(t)Bu(t)=0,qquad tgeq0 end{equation*} where the operator $-A$ is a self-adjoint accretive operator on a Hilbert space and $p(cdot)$ is the control function. The proof is based on a linearization argument. We prove that the linearized system is exacly controllable and we apply the moment method to build a control $p(cdot)$ that steers the solution to the ground state in finite time. Finally, we use such a control to bring the solution of the nonlinear equation arbitrarily close to the ground state solution with doubly exponential rate of convergence. We give several applications of our result to different kinds of parabolic equations.