Abstract
This paper studies a periodic optimal control problem governed by a one-dimensional system, linear with respect to the control \begin{document}$ u $\end{document} , under an integral constraint on \begin{document}$ u $\end{document} . We give conditions for which the value of the cost function at steady state with a constant control \begin{document}$ \bar u $\end{document} can be improved by considering periodic control \begin{document}$ u $\end{document} with average value equal to \begin{document}$ \bar u $\end{document} . This leads to the so-called over-yielding met in several applications. With the use of the Pontryagin Maximum Principle, we provide the optimal synthesis of periodic strategies under the integral constraint. The results are illustrated on a single population model in order to study the effect of periodic inputs on the utility of the stock of resource.
Highlights
In many applications, the control of dynamical models allows to drive the state x of a system to an operating point, typically a steady state xwhich is an equilibrium point of the dynamics under a constant control u
From a mathematical view point, the integral constraint on the input brings two main difficulties: 1. the existence of non-constant periodic trajectories with a control satisfying the integral constraint, 2. the characterization of an optimal control under both constraints of periodicity of the trajectory and the integral constraint on the input, that we propose to tackle here for scalar dynamics in general framework
We have shown that under concavity assumptions, the optimal trajectory is the steady-state solution, that is, no over-yielding is possible
Summary
The control of dynamical models allows to drive the state x of a system to an operating point, typically a steady state xwhich is an equilibrium point of the dynamics under a constant control u. Since (H3) implies (H ), Proposition 3.2 gives the uniqueness of a T -admissible B+B−B+ trajectory (see Remark 6), which amounts to state that there are exactly two extremals with two switches (corresponding to n = 1), given by the controls uT (·) and uT (·) Recall that they have same cost because uT (·) is obtained by a time translation of uT (·). There could exist extremal trajectories taking values outside the interval [xm, xM ], without requiring additional assumption on the function ψ outside this set For this purpose, we consider the two controls u− and u+ defined by one switching time t− ∈ (0, T ) (for u−) and t+ ∈ (0, T ) (for u+) as u−(t) =. We distinguish two cases depending if Emax is below or above E
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