Free time optimal control problems with time delays

Abstract

This paper provides necessary conditions of optimality for optimal control problems with time delays in both state and control variables. Different versions of the necessary conditions cover fixed end-time problems and, under additional hypotheses, free end-time problems. The conditions improve on previous available conditions in a number of respects. They can be regarded as the first generalized Pontryagin Maximum Principle for fully non-smooth optimal control problems, involving delays in state and control variables, only special cases of which have previously been derived. Even when the data is smooth, the conditions advance the existing theory. For example, we provide a new 'two-sided' generalized transversality condition, associated with the optimal end-time, which gives more information about the optimal end-time than the 'one-sided' condition in the earlier literature. But there are improvements in other respects, relating to the treatment of initial data, specifying past histories of the state and control, and to the unrestrictive nature of the hypotheses under which the necessary conditions are derived. Nonsmoothness:We provide the first set of necessary conditions, in the form of a generalized Pontryagin Maximum Principle, for 'fully' nonsmooth problems (i.e. problems in which the only regularity hypothesis on the data w.r.t. the state variable is 'Lipschitz continuity') involving delays in states and controls. They resemble the classical necessary conditions for 'smooth' problems except that, in the costate relation (1.3) classical derivatives are replaced by set-valued subdifferentials of nonsmooth analysis. Two earlier papers [6], [7] provide necessary conditions for nonsmooth optimal control problems with time delays in the state alone. The difference is that, in these papers, the dynamic constraint is modelled as a differential inclusion, and the relation for the costate arc (combined with the Weierstrass condition) is a generalization of Clarke's Hamiltonian inclusion condition with 'advanced' arguments. As observed in [7, Section 1,], necessary conditions expressed in terms of the Hamiltonian inclusion imply the nonsmooth Maximum Principle only for problems having special structure and not 'fully' nonsmooth problems, as in this paper. Furthermore, the methods of [6] and [7] cannot be adapted to cover problems with time delays in the control, because it is not possible to express a controlled delay differential equation (with delays in the control) as a delay differential inclusion (which can take account only of delays in the state).[7] allows both distributed and discrete delays (in the state variable), whereas we allow only discrete delays (in both state and control variables). Necessary conditions for optimal control problems involving differential inclusions are also provided in [3] and [15] for fixed time optimal control problems involving a single time delay in the state. We mention that Warga [19] showed that a broad class of optimal control problems involving delays and/or functional diff...

A sufficient optimality condition for delayed state-linear optimal control problems

“…An illustrative example. Let us consider problem (P ) given by 4]. Thus, we have that n = m = 1, a = 0, b = 4, r = 2, s = 1, f 0 (t, x(t), x(t − 2)) = x(t), g 0 (t, u(t), u(t − 1)) = 100u 2 (t), A(t) = A D (t) = 1, g(t, u(t)) = 0 and g D (t, u(t − 1)) = −10u(t − 1).…”

“…Furthermore, necessary conditions and Hamilton-Jacobi equations are derived. In 2013, Boccia, Falugi, Maurer and Vinter derived necessary conditions for a free end-time optimal control problem subject to a non-linear differential system with multiple delays in the state [4]. The control variable is not influenced by time lags in [4].…”

“…In 2013, Boccia, Falugi, Maurer and Vinter derived necessary conditions for a free end-time optimal control problem subject to a non-linear differential system with multiple delays in the state [4]. The control variable is not influenced by time lags in [4]. Recently, in 2017, Boccia and Vinter obtained necessary conditions for a fixed end-time problem with a constant and unique delay for all variables, as well as free end-time problems without control delays [5].…”

A computational method for solving time-delay optimal control problems with free terminal time

“…Any pair (T,ũ) ∈ T ×Ũ is called an admissible pair. Letx(·|T,ũ) denote the solution of system (7) corresponding to a given admissible pair (T,ũ) ∈ T ×Ũ. Then the canonical constraints (4) becomẽ…”

“…We are only aware of one reference (reference [7]) that tackles this class of problems. This reference describes a two-stage optimization approach in which the terminal time is optimized in the outer stage, and the control function is optimized in the inner stage.…”