An Introduction to Mathematical optimal control

Journal of Mathematical Sciences, 1999

Here ~ = ~,a~ t E N1 is time, and x = (xl, 9-. , z ~) is the phase variable, which assumes values in the ndimensional arithmetic space l~ ~. The vector u = (ul, ... ,u'~), which assumes values in the space 1~ '~, is called the control parameter, or briefly, the control. It is expressed as u = (ul, u2), where ul denotes the first rnl coordinates of the vector u, and u2 represents the remaining rn2 coordinates. Here ml + m2 = m. The reason for subdividing the control parameter into two components ul and u2 will be discussed below. The functions R, G, K1, and /(2 take values in the arithmetic spaces of dimension d(R), d(G), d(K~), and d(K2), respectively. Here and in what follows, d(z) denotes the dimension of the finite-dimensional vector z. The coordinates of these vectorivalued functions are denoted by superscripts R j, G j, etc. The functions K0 and f0 are scalar-valued functions, whereas f is an n-dimensional vector-valued function. The functions G,/to, K1, and/(2 are assumed to be continuously differentiable. The functions fo, f, and R are assumed to be continuously differentiable with respect to (x, ul) for almost all t, measurable in t for every fixed (x,u), together with their partial derivatives with respect to (x, ul). Furthermore, we assume that these functions and their partial derivatives with respect to (z, ua) are bounded and continuous in (x, u) uniformly with respect to x, u, and t on any bounded set. As the class of admissible controls, we take the set of all possible functions u(-) = (ul('), u~(.)), which are defined, measurable, and essentially bounded, each on its own finite interval, and satisfy the condition u2(t) E U2(t) for almost all t. Here U2 is a given multivalued ,mapping which associates every t with a nonempty subset U2(t) C_ Nm~. A measurable function v is said to be a measurable selector of a multivalued map U2 if v(t) e U2(t)~/t.

2007

Optimization Letters, 2012

Journal of Applied Mathematics and Mechanics, 1992

SIAM Journal on Control and Optimization, 1991

The authors provide several characterizations of optimal trajectories for the classical Meyer problem arising in optimal control. For this purpose they study the regularity of directional derivatives of the value function: for instance it is shown that for smooth control systems the value function V is continuously differentiable along an optimal trajectory x : [to, 1 1 + Rn provided V is differentiable a t the initial point (to,x(tO)). Then they deduce the upper semicontinuity of the optimal feedback map and address the problem of optimal design, obtaining sufficient conditions for optimality. Finally it is shown that the optimal control problem may be reduced t o a viability problem. Alexander B. Kurzhanski Chairman System and Decision Sciences Program Contents Value function in optimal control Some preliminaries on nonsmooth functions Necessary and sufficient conditions for optimality 4 Semiconcavity properties of the value function Optimal feedback Viability approach to optimal control Problem with end point constraints Some characterizations of optimal trajectories in control theory