Aaron Strauss

AN INTRODUCTION TO OPTIMAL CONTROL THEORY.

Abstract : The report presents an introduction to some of the concepts and results currently popular in optimal control theory. The introduction is intended for someone acquainted with ordinary differential equations and real variables, but with no prior knowledge of control theory. The material covered includes the problems of controllability, controllability using special (e.g., bang-bang) controls, the geometry of linear time optimal processes, general existence of optimal controls, and the Pontryagin maximum principle. (Author)

Perturbation theorems for ordinary differential equations

x’ =.f(4 4, (NJ x’ =“f@, 4 + g(4 4, (P) x’ =.f(4 4 + g(c 4 + h(t), (PI) where f(t, X) is continuous, satisfies a Lipschitz condition on some semicylinder, f(t, 0) = 0, and x = 0 is uniform asymptotically stable for (N). Let g(t, x) and h(t) be sufficiently smooth for local existence and uniqueness. Consider the conditions (Hi): There exists r > 0 such that if 1 x 1 < Y, then 1 g(t, x)1 < y(t) for all t 3 0, where

Perturbing uniform asymptotically stable nonlinear systems

and that f and g satisfy certain conditions. We always assume that f and g are at least continuous from [0, CO) x R” to Rd. Assume temporarily that the solutions of (P) are unique but do not assume that the zero function is a solution of (P). In fact EvUAS is a natural generalization of uniform asymptotic stability in which it is not assumed that the zero function is a solution (Lemma 2.7). One main result (see Theorems 4.4, 5.2, 6.1 and 7.1) is

ON A PERTURBED VOLTERRA INTEGRAL EQUATION.

Abstract : For the Volterra integral equation x(t) = f(t) - the integral from 0 to t of (a(t,s)(x(s) + g(s,x(s))) ds), if the resolvent kernel of a(t,s) is sufficiently well-behaved, and if the absolute value of g(t,s) approaches 0 as t approaches infinity in some sense, then the absolute value of (x(t) - y(t)) approaches 0 as t approaches infinity, where y(t) is the solution of y(t) = f(t) - the integral from 0 to t of (a(t,s) y(s) ds). (Author)

Asymptotic behavior for differential equations which cannot be locally linearized

For functions ƒ which are continuous and locally Lipschitz the authors define a multi-valued differential Df and prove that if all solutions of the multi-valued differential equation u′ ϵ Df(u) approach zero as t → ∞, then all solutions x(·) of x′ = ƒ(x) with small |x(0)| approach zero exponentially as t → ∞. If ƒ is continuously differentiable, then Df coincides with the (single-valued) Frechet differential of ƒ. Other results on the asymptotic behavior of solutions of perturbed, multi-valued differential equations are presented.

Two countable systems of differential inequalities with applications to the stability of linear quadratic differential games

Abstract This paper proves that the feedback controls synthesizing the equilibrium solutions to linear quadratic differential games possess certain stability characteristics.

Infinite systems of differential inequalities defined recursively

Abstract It is shown that all solutions of certain recursively defined, infinite systems of differential inequalities converge to zero. Convergence is global for initial value inequalities, but only local for boundary value inequalities. Applications to the stability of differential games are sketched.

Existence Theorems for Optimal Control Problems

In this chapter we discuss sufficient conditions for the existence of an optimal control for the general problem:

Linear Autonomous Time-Optimal Control Problems

\dot x = f\left( {t,x,u} \right),x\left( 0 \right) = {x_0},u\left( \cdot \right) \in {U_m},