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

Abstract

We give answer to an open question by proving a sufficient optimality condition for state-linear optimal control problems with time delays in state and control variables. In the proof of our main result, we transform a delayed state-linear optimal control problem to an equivalent non-delayed problem. This allows us to use a well-known theorem that ensures a sufficient optimality condition for non-delayed state-linear optimal control problems. An example is given in order to illustrate the obtained result.2010 Mathematics Subject Classification. Primary: 49K15; Secondary: 34H99.Optimal control problems with a differential system that is linear both in state and control variables have been studied in [7,9,10,12,26,28,29,31,35,37]. In [10,28,37], the system is delayed with respect to state and control variables. In [9,35], the system only considers delays in the state variable. Chyung and Lee derive necessary and sufficient optimality conditions in [9] while Oǧuztöreli only proves necessary conditions [35]. Certain necessary conditions analysed by Chyung and Lee in [9] have been already derived in [23,40,41]. However, the system considered in [9] is different from the previously studied hereditary systems, which do not require a initial function of state. In [12], Eller et al. derive a sufficient condition for a control to be optimal for certain problems with time delay. The problems studied by Eller et al. and Khellat, respectively in [12] and [26], consider only one constant lag in the state. The research done by Lee in [31] is different from ours, because in [31] the aim is to minimize a cost functional, which does not consider delays, subject to a differential system that is linear in state and control variables, and to another constraint. In their differential system, the state variable depends on a constant and fixed delay and the control variable depends on a constant lag, which is not specified a priori. Note that the differential system of the problem considered in [29] is similar to the one of [31]. Although Banks has studied non-linear delayed problems without lags in the control, he has also analyzed problems that are linear and delayed with respect to control [2]. Recently, Cacace et al. studied optimal control problems that involve linear differential systems with variable delays only in the control [7]. The problems analyzed in the present paper are different from those considered in the mentioned works, because here the problems involve differential systems that are linear with respect to state, but not with respect to the control. Furthermore, we consider a constant lag in the state and another one in the control. These two delays are in general not equal.In [20], Hughes firstly consider variational problems with only one constant lag and derive various necessary and a sufficient optimality condition for them. The variational problems in [20] can easily be transformed to control problems with only one constant delay (see, e.g., [34, p. 53-54]). Hughes also investigate an optimality condition for a ...

Stability and optimal control of a delayed HIV/AIDS-PrEP model

“…Although, recently sufficient optimality conditions for non-linear optimal control problems with state and control delays have been proved, see e.g. [14], these conditions are difficult to apply in the proposed problem.…”

“…We propose the optimal control problem that consists to find the optimal strategy for PrEP implementation ũ(•) ∈ L 1 ([0, t f ], R) and the associated optimal state trajectories x(•), satisfying the delayed control system (11), the initial conditions (12) and where the control ũ ∈ Ω minimizes the objective functional (14).…”

“…The costs associated to the number of individuals with HIV infection I and with the effective distribution of PrEP can assume different values, depending on the region under consideration. In this Subsection, we assume different values for the weight constants w i , i = 1, 2, considered in the cost functional (14). We consider three cases:…”