Abstract
Two sufficiency theorems for parametric and a nonparametric problems of Bolza in optimal control are derived. The dynamics of the problems are nonlinear, the initial and final states are free, and the main results can be applied when nonlinear mixed time-state-control inequality and equality constraints are presented. The deviation between admissible costs and optimal costs around the optimal control is estimated by functionals playing the role of the square of some norms.
Highlights
In this paper, we derive two new sufficiency theorems in optimal control problems as the parametric and nonparametric problems of Bolza with nonlinear dynamics, free initial and final states, and inequality and equality mixed time-state-control constraints
The fundamental components of the sufficiency theorems of this article are a similar version of the Pontryagin maximum principle, a hypothesis usually called the transversality condition, a crucial second order inequality arising from the original algorithm employed to prove one of the sufficiency theorems, a related hypothesis of the Legendre–Clebsh necessary condition, the positivity of a quadratic function on the cone of critical directions, and a fundamental integral Weierstrass inequality involving a function whose role is parallel to the Hamiltonian of the problem
The optimal control of the proposed optimal process need not be continuous but only measurable, see, for example, [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17], where the authors study several optimal control problems having a degree of generality very similar to the one treated in this paper and where the continuity of the optimal controls is a crucial assumption in those sufficiency theories
Summary
We derive two new sufficiency theorems in optimal control problems as the parametric and nonparametric problems of Bolza with nonlinear dynamics, free initial and final states, and inequality and equality mixed time-state-control constraints. We shall be concerned with the nonparametric optimal control problem, denoted by (P), of minimizing the functional. K) be functions having first and second continuous derivatives with respect to x and u on T × Rn × Rm, and consider the isoperimetric nonparametric optimal control problem of minimizing subject to x(ti) ∈ Bi for i = 0, 1. Example 1 below shows how Corollary 1 can be applied In the former, an inequalityequality constrained optimal control problem is solved by verifying that the first order sufficiency conditions ρ(t) = −Hx∗(t, x0(t), u0(t), ρ(t), μ(t)) (a.e. in T), Hu∗(t, x0(t), u0(t), ρ(t), μ(t)) = 0 (t ∈ T), are satisfied by an element (x0, u0, ρ, μ). It is worth mentioning that the optimal control model mentioned above saved lives and minimized the economical costs of the pharmaceutical interventions
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