A hybrid approach to optimal control problems with bounded state - Miele, A.; Tietze, J.L.; Cloutier, J.R. (Computers & mathematics with applications (1987), 1975)
This paper considers the numerical solution of optimal control problems involving a functional I subject to differential constraints, a state inequality constraint, and terminal constraints. The problem is to find the state x(t), the control u(t), and the parameter π so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy. The approach taken is a sequence of two-phase processes or cycles, each composed of a gradient phase and a restoration phase. The gradient phase involves a single iteration and is designed to decrease the functional, while the constraints are satisfied to first order. The restoration phase involves one or several iterations and is designed to restore the constraints to a predetermined accuracy, while the norm of the variations of the control and the parameter is minimized. The principal property of the algorithm is that it produces a sequence of feasible suboptimal solutions: the functions x(t), u(t), π obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the functionals of any two elements of the sequence are comparable. The technique employed is of the hybrid type, in an attempt to combine some of the best features of both the indirect and direct approaches. While a predetermined number and sequence of subarcs are assumed (a feature of direct methods), enforcement of the state inequality constraint is obtained through a Valentine-type representation (a feature of indirect methods). By properly choosing the analytical form of certain non-differential constraints to be satisfied by the augmented control along each subarc composing the extremal arc (these nondifferential constraints are arrived at through the Valentine-type transformation), one ensures satisfaction of the state inequality constraint everywhere. Specifically, strict inequality is enforced for the subarcs internal to the state boundary and strict equality is enforced for the subarcs lying on the state boundary. To facilitate the numerical solution on digital computers, the actual time θ is replaced by the normalized time t, defined in such a way that each subarc composing the extremal arc has a normalized time length Δ t = 1. In this way, variable-time corner conditions and variable-time terminal conditions are transformed into fixed-time corner conditions and fixed-time terminal conditions. The actual times θ 1, θ 2, τ at which (i) the state boundary is entered, (ii) the state boundary is exited, and (iii) the terminal manifold is reached are regarded to be components of the parameter π being optimized. The numerical examples illustrating the theory demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.