Hamilton-Jacobi Inequality Research Articles

A necessary condition for nonlinear [formula omitted] control via measurement feedback

In a series of recent papers, it was shown that the solution of the problem of disturbance attenuation for nonlinear system is related to the existence of solutions of a pair of Hamilton-Jacobi inequalities in n independent variables, associated with state-feedback and output-injection design. The purpose of the present paper is to discuss the necessity of the existence of solutions of the Hamilton-Jacobi inequality which determines the output-injection gain.

125 Nonlinear control and Hamilton-Jacobi inequalities

125 Nonlinear control and Hamilton-Jacobi inequalities

Nonlinear H ∞ Control and Hamilton-Jacobi Inequalities

This chapter describes the nonlinear H ∞ control and Hamilton–Jacobi inequalities. Although the linear H ∞ control problem was originally formulated in the frequency domain, the translation of it to the time-domain has a clear interpretation that naturally extends to nonlinear systems as well. The H ∞ optimal control problem can be formulated as the optimal attenuation of the L 2 -gain from unknown disturbances entering the system to a set of to-be-controlled variables. As a consequence, H ∞ control can be centered on the classical Bounded Real Lemma in the linear case, and in the nonlinear case to generalizations of this lemma as developed by Willems and Hill-Moylan. Nonlinear H ∞ control appears to be an extremely rich area, where modern nonlinear control theory can be fruitfully combined with more classical theory.

The generalised Hamilton-Jacobi inequality and the stability of equilibria in nonlinear elasticity

The generalised Hamilton-Jacobi inequality and the stability of equilibria in nonlinear elasticity

Optimal trajectories associated with a solution of the contingent Hamilton-Jacobi equation

In this paper we study the existence of optimal trajectories associated with a generalized solution to the Hamilton-Jacobi-Bellman equation arising in optimal control. In general, we cannot expect such solutions to be differentiable. But, in a way analogous to the use of distributions in PDE, we replace the usual derivatives with “contingent epiderivatives” and the Hamilton-Jacobi equation by two “contingent Hamilton-Jacobi inequalities.” We show that the value function of an optimal control problem verifies these “contingent inequalities.” Our approach allows the following three results: (a) The upper semicontinuous solutions to contingent inequalities are monotone along the trajectories of the dynamical system. (b) With every continuous solutionV of the contingent inequalities, we can associate an optimal trajectory along whichV is constant. (c) For such solutions, we can construct optimal trajectories through the corresponding optimal feedback. They are also “viscosity solutions” of a Hamilton-Jacobi equation. Finally, we prove a relationship between superdifferentials of solutions introduced by Crandallet al. [10] and the Pontryagin principle and discuss the link of viscosity solutions with Clarke's approach to the Hamilton-Jacobi equation.

A modified Hamilton-Jacobi approach in the generalized problem of Bolza

We are concerned with developing sufficiency criteria for the generalized problem of Bolza, where the Hamiltonian is neither concave-convex nor differentiable. The core of the approach is the “modified” Hamilton-Jacobi inequality which leads to a new type of sufficient conditions. This latter is then used to derive new first order and known second order sufficient conditions.