Proximal point method for optimal control processes governed by ordinary differential equations

Abstract

This paper is concerned with the proximal-based approach to linear and finite-difference approximations of constrained convex optimal control problems. We consider control systems governed by ordinary differential equations in the presence of additional terminal/state inequalities and propose a numerical method derived from the proximal point algorithm. The aim of the paper is to study the convergence properties of the obtained conceptual algorithm and to show that it can be used to compute approximate optimal controls.

A Proximal Point Based Approach to Optimal Control of Affine Switched Systems

“…We refer to Attia et al (2005Attia et al ( , 2010, , Azhmyakov and Noriega Morales (2010), Branicky et al (1998), Cassandras et al (2001), Egerstedt et al (2006), Garavello and Piccoli (2005), Liberzon (2003), Lygeros (2003, Moor and Raisch (1999) and Shaikh andCaines (2007), Sussmann (1999) for some concrete definitions and concepts of hybrid/switched control systems. Note that for some classes of the above systems the possible switching mechanisms are formalized by different types of the characteristic functions β.…”

“…The same is true in connection with various numerical solution schemes associated with OCPs. We refer to (see e.g., Azhmyakov and Noriega Morales 2010;Büskens and Maurer 2000;Cassandras et al 2001;Maurer 1976;Polak 1997;Sakawa et al 1981;Teo et al 1991) for theoretical and some computational aspects. The necessary optimality conditions for convex minimization problems (3) and (4) are also sufficient.…”

“…Moreover, we examine the above optimization methods in combination with a regularization scheme, namely, in combination with proximal point technique. This combination makes it possible to obtain a numerically stable general algorithm (see Azhmyakov and Noriega Morales 2010;Kaplan and Tichatschke 1994 for details).…”

Primal-Dual Extragradient Methods for Nonlinear Nonsmooth PDE-Constrained Optimization

“…Consequently, they recently have received increasing interest also in the context of optimal control; see, e.g., [15,13]. In addition, other proximal point methods for optimal control problems have been treated in [1] and [18]; in particular, the latter is concerned with classical forward-backward splitting for sparse control of linear elliptic PDEs. However, so far these methods have only been considered in the finite-dimensional setting, i.e., after discretizing (1.1), or for specific (linear) problems.…”