Marc Quincampoix

Impulse differential inclusions: a viability approach to hybrid systems

Impulse differential inclusions are introduced as a framework for modeling hybrid phenomena. Connections to standard problems in the area of hybrid systems are discussed. Conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an impulse differential inclusion. For sets that violate these conditions, methods are developed for approximating their viability and invariance kernels, that is the largest subset that is viable or invariant under the action of the impulse differential inclusion. The results are demonstrated on examples.

Generalized neural network for nonsmooth nonlinear programming problems

In 1988 Kennedy and Chua introduced the dynamical canonical nonlinear programming circuit (NPC) to solve in real time nonlinear programming problems where the objective function and the constraints are smooth (twice continuously differentiable) functions. In this paper, a generalized circuit is introduced (G-NPC), which is aimed at solving in real time a much wider class of nonsmooth nonlinear programming problems where the objective function and the constraints are assumed to satisfy only the weak condition of being regular functions. G-NPC, which derives from a natural extension of NPC, has a neural-like architecture and also features the presence of constraint neurons modeled by ideal diodes with infinite slope in the conducting region. By using the Clarkes generalized gradient of the involved functions, G-NPC is shown to obey a gradient system of differential inclusions, and its dynamical behavior and optimization capabilities, both for convex and nonconvex problems, are rigorously analyzed in the framework of nonsmooth analysis and the theory of differential inclusions. In the special important case of linear and quadratic programming problems, salient dynamical features of G-NPC, namely the presence of sliding modes , trajectory convergence in finite time, and the ability to compute the exact optimal solution of the problem being modeled, are uncovered and explained in the developed analytical framework.

Optimal times for constrained nonlinear control problems without local controllability

We study optimal times to reach a given closed target for controlled systems with a state constraint. Our goal is to characterize these optimal time functions in such a way that it is possible to compute them numerically and we do not need to compute trajectories of the controlled system. In this paper we provide new results using viability theory. This allows us to study optimal time functions free from the controllability assumptions classically made in the partial differential equations approach.

Convergence of Neural Networks for Programming Problems via a Nonsmooth Lojasiewicz Inequality

This paper considers a class of neural networks (NNs) for solving linear programming (LP) problems, convex quadratic programming (QP) problems, and nonconvex QP problems where an indefinite quadratic objective function is subject to a set of affine constraints. The NNs are characterized by constraint neurons modeled by ideal diodes with vertical segments in their characteristic, which enable to implement an exact penalty method. A new method is exploited to address convergence of trajectories, which is based on a nonsmooth Lstrokojasiewicz inequality for the generalized gradient vector field describing the NN dynamics. The method permits to prove that each forward trajectory of the NN has finite length, and as a consequence it converges toward a singleton. Furthermore, by means of a quantitative evaluation of the Lstrokojasiewicz exponent at the equilibrium points, the following results on convergence rate of trajectories are established: 1) for nonconvex QP problems, each trajectory is either exponentially convergent, or convergent in finite time, toward a singleton belonging to the set of constrained critical points; 2) for convex QP problems, the same result as in 1) holds; moreover, the singleton belongs to the set of global minimizers; and 3) for LP problems, each trajectory converges in finite time to a singleton belonging to the set of global minimizers. These results, which improve previous results obtained via the Lyapunov approach, are true independently of the nature of the set of equilibrium points, and in particular they hold even when the NN possesses infinitely many nonisolated equilibrium points

Linear Programming Approach to Deterministic Infinite Horizon Optimal Control Problems with Discounting

We investigate relationships between the deterministic infinite time horizon optimal control problem with discounting, in which the state trajectories remain in a given compact set

On the Reachability Problem for Uncertain Hybrid Systems

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Differential inclusions and target problems

, and a certain infinite dimensional linear programming (IDLP) problem. We introduce the problem dual with respect to this IDLP problem and obtain some duality results. We construct necessary and sufficient optimality conditions for the optimal control problem under consideration, and we give a characterization of the viability kernel of

Pursuit Differential Games with State Constraints

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Existence of stochastic control under state constraints

. We also indicate how one can use finite dimensional approximations of the IDLP problem and its dual for construction of near optimal feedback controls. The construction is illustrated with a numerical example.

DETERMINISTIC DIFFERENTIAL GAMES UNDER PROBABILITY KNOWLEDGE OF INITIAL CONDITION

In this paper, we revisit the problem of designing controllers to meet safety specifications for hybrid systems, whose evolution is affected by both control and disturbance inputs. The problem is formulated as a dynamic game and an appropriate notion of hybrid strategy for the control inputs is developed. The design of hybrid strategies to meet safety specifications is based on an iteration of alternating discrete and continuous safety calculations. We show that, under certain assumptions, the iteration converges to a fixed point, which turns out to be the maximal set of states for which the safety specifications can be met. The continuous part of the calculation relies on the computation of the set of winning states for one player in a two player, two target, pursuit evasion differential game. We develop a characterization of these winning states (as well as the winning states for the other player for completeness) using methods from nonsmooth analysis and viability theory.