Convexity, convergence and feedback in optimal control

Abstract

The results of this thesis are oriented towards the study of convex problems of optimal control in the extended piecewise linear-quadratic format. Such format greatly extends the classical linear-quadratic regulator problem and allows for the treatment of control constraints, including state-dependent ones. The Hamiltonian system associated with a control problem, the optimal feedback mapping, and the value function are objects of main interest. Several tools of nonsmooth and convex analysis are developed, including a new approximation scheme for convex functions, characterizations of a saddle function through the properties of it's conjugate, and a new distance formula for monotone operators. The optimal feedback mapping for control problems is given, in terms of subdifferentials of the corresponding Hamiltonian and of the value function. The Hamiltonian system is employed to investigate the regularity properties of the value function for the problem in question. Conditions for differentiability of the value function and single-valuedness of the feedback in an extended linear-quadratic control problem are stated, in terms of the matrices and constraint sets defining the problem. Application of convex analysis to differential games yields explicit formulas for equilibrium controls and a generalized Hamiltonian equation describing an equilibrium trajectory.