A duality scheme for convex control problems with time-delay
The paper concerns optimal control problems with time-delay, both on the cost and on the dynamics, of the type \[ \text{ minimize } \;J(x,u):= l(x(0),x(1)) + \int_0^1 K(t,x(t),u(t),u(t-\theta)) dt \] subject to \(\dot x (t) = A_0(t)x(t) + A_1(t)x(t-\tau) + B(t)u(t)\), where the delay parameters \(\theta, \tau\) can be different and the functions \(l(\cdot,\cdot)\), \(k(t,\cdot,\cdot,\cdot)\) are convex. A Fenchel duality scheme is introduced in order to formulate a dual problem having a similar character to the primal one. The solutions of the dual problem are interpreted as Lagrange multipliers for the dynamic constraints on the primal problem. A necessary and sufficient optimality condition of a subdifferential type is established. This technique allows to improve by some aspects analogous results obtained by the elimination of the time-delay, based on the method of steps [\textit{S. C. Huang}, J. Optimization Theory Appl. 3, 316-360 (1969; Zbl 0172.12904)] or on the introduction of an infinite-dimensional state space [\textit{W. L. Chan} and \textit{S. P. Yung}, IMA J. Math. Control Inf. 4, 251-262 (1987; Zbl 0629.49008)].
scientific article; zbMATH DE number 1246323
scientific article