Geometric Control and Nonsmooth Analysis

The following sections are included:

• PREFACE

• CONFERENCE COMMITTEES

• CONTENTS

The paper is devoted to singular perturbation problems with a finite number of scales where both the dynamics and the costs may oscillate. Under some coercivity assumptions on the Hamiltonian, we prove that the value functions converge locally uniformly to the solution of an effective Cauchy problem for a limit Hamilton-Jacobi equation and that the effective operators preserve several properties of the starting ones; under some additional hypotheses, their explicit formulas are exhibited. In some special cases we also describe the effective dynamics and costs of the limiting control problem. An important application is the homogenization of Hamilton-Jacobi equations with a finite number of scales and a coercive Hamiltonian.

This paper provides a survey of the theory of patchy feedbacks, and its applications to asymptotic stabilization and optimal control. It also contains two new results, showing the robustness of sub-optimal patchy feedbacks both in the case of (internal and external) deterministic disturbances, and of random perturbations modelled by stochastic Brownian motion.

Measure-valued coefficients arise when limits of rapidly varying parameters of control systems are examined. The rapidly varying parameter is considered then a perturbation of a measure-valued parameter. We provide quantitative estimates for the sensitivity of the optimal value and near optimal solutions, with respect to such perturbations. As a particular case, which is of interest for its own sake, we establish quantitative estimates for the sensitivity of ordinary control systems with respect to relaxed controls.

We introduce a class of difference/differential equations which is sufficiently large to include systems of interest for applications, but at the same time sufficiently easy to handle. In this framework, we give in particular a detailed and rather complete study of the asymptotic behavior of pairs of oscillators. We finally introduce appropriate notions of stability and extend Liapunov first and second theorem.

In this paper we review some recent results on stability of multilinear switched systems, under arbitrary switchings. An open problems is stated

The Maximum principle in control theory provides necessary optimality conditions for a given trajectory in terms of the co-state, which is the solution of a suitable adjoint system. For constrained problems the adjoint system contains a measure supported at the boundary of the constraint set. In this paper we give a representation formula for such a measure for smooth constraint sets and nice Hamiltonians. As an application, we obtain a perimeter estimate for constrained attainable sets.

In this paper we discuss the validity of the Hopf-Lax representation formula for solutions of evolution Hamilton- Jacobi equations governed by state-dependent and perhaps non coercive Hamiltonians. Some applications to PDE's on the Heisenberg group and finite differences approximations are also presented.

We review the long-standing issue of regularity of solutions to the basic problem in the calculus of variations, in both the one-dimensional and the multidimensional settings. It is shown how certain recent results fit in with the classical ones, in particular the theories of De Giorgi and Hilbert-Haar.

In this paper we announce a new framework for a rigorous stability analysis of sliding-mode controllers. We give unrestrictive conditions under which such feedback controllers are robustly stabilizing. These conditions make allowance for large disturbance signals, for modeling, actuator and observation measurement errors, and also for the effects of digital implementation of the control. The proposed stability analysis techniques involve two Lyapunov-type functions. The first is associated with passage to the sliding surface in finite time; the second, with convergence to the state associated with the desired equilibrium point. Application of the techniques is illustrated with reference to higher-order linear systems in control canonical form.

We consider the problem of weak differentiability for a parameterized family of trajectories. The study is motivated by various problems in ordinary differential equations and control theory.

Some results are presented and their applicability is shown by means of few examples. The methods are based on generalized differentiation of flows or, more generally, of set-valued maps.

We investigate the sampled-data redesign problem for nonlinear control affine multi-input systems and consider sampled-data feedback laws for which the trajectories of the sampled-data closed loop system converge to the continuous time trajectories with a prescribed rate of convergence as sampling time vanishes. We analyze geometric conditions for the existence of such sampled-data feedback laws and give formulae and algorithms for their computation.

In this paper we consider control-affine systems driven by controls that are not absolutely continuous, so that a solution in the classical sense is not defined. We discuss the possibility of defining a generalized notion of solution. Focusing on the Heisenberg example, we state some conjectures that would justify an extension of the definition of solution, and we analyze them by means of examples and counterexamples.

We characterize the state constrained bilateral minimal time function as the unique proximal solution of a partial Hamilton-Jacobi equations with certain boundary conditions. This generalizes [13, Theorem 3.4] where Stern studied the unilateral case.

In this article we consider the generalized problem of Bolza with different time varying delays appearing in the state and velocity variables. We derive existence of solutions and prove a natural extension of a decoupling theorem, which was originally introduced by Clarke in the non delay case² and was previously extended⁷ to the neutral case when both of the delays appearing are given by one function.

We present the stabilization problem in the context of nonholonomic control systems and nonholonomic distributions. Then, we introduce the notions of smooth repulsive stabilizing feedbacks and sections, and review recent results on the existence of such objects.

The graph of the reachable set is characterized via limiting Hamilton-Jacobi inequalities when the system dynamics satisfy a discontinuous dissipative Lipschitz condition.

An optimal control problem with linear dynamics and convex but not necessarily quadratic and possibly infinite-valued or nonsmooth costs can be analyzed in an appropriately formulated duality framework. The paper presents key elements of such a framework, including a construction of a dual optimal control problem, optimality conditions in open loop and feedback forms, and a relationship between primal and dual optimal values. Some results on differentiability and local Lipschitz continuity of the gradient of the optimal value function associated with a convex optimal control problem are included.

The paper proves second order sufficient conditions for the strong optimality of singular extremals of the first kind. The conditions are given both in Hamiltonian formulation and by means of the coercivity of a suitable coordinate-free second variation. The conditions are close to the necessary ones in the usual sense, namely we require strict inequalities where the necessary conditions have mild inequalities.

In a series of nonsmooth versions of the Pontryagin Maximum Principle, we used generalized differentials of set-valued maps, flows, and abstract variations. Bianchini and Stefani have introduced a notion of possibly high-order variational vector that has the summability property. We consider a slightly more general class of variational vectors than that defined by Bianchini and Stefani, and prove that the convex combinations of these vectors arise as “differentials” of variations that are differentiable in the sense of one of our generalized differentiation theories, namely, that of “approximate generalized differential quotients” (AGDQs).

The following sections are included:

• LIST OF PARTICIPANTS

• AUTHOR INDEX