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In this paper, we develop several necessary conditions of turnpike property for generalizaid linear-quadratic (LQ) optimal control problem in infinite dimensional setting. The term 'generalized' here means that both quadratic and linear terms are considered in the running cost. The turnpike property reflects the fact that over a sufficiently large time horizon, the optimal trajectories and optimal controls stay for most of the time close to a steady state of the system. We show that the turnpike property is strongly connected to certain system theoretical properties of the control system. We provide suitable conditions to characterize the turnpike property in terms of the detectability and stabilizability of the system. Subsequently, we show the equivalence between the exponential turnpike property for generalized LQ and LQ optimal control problems.

In this paper we investigate the turnpike property for constrained LQ optimal control problem in connection with dissipativity of the control system. We determine sufficient conditions to ensure the turnpike property in the case of a turnpike reference possibly occurring on the boundary of the state constraint set.

In this paper we introduce an optimal control approach to Richards' equation in an irrigation framework, aimed at minimizing water consumption while maximizing root water uptake. We first describe the physics of the nonlinear model under consideration, and then develop the first-order necessary optimality conditions of the associated boundary control problem. We show that our model provides a promising framework to support optimized irrigation strategies, thus facing water scarcity in irrigation. The characterization of the optimal control in terms of a suitable relation with the adjoint state of the optimality conditions is then used to develop numerical simulations on different hydrological settings, that supports the analytical findings of the paper.

We deduce a sufficient condition of the exponential (integral) turnpike property for infinite dimensional generalized linear-quadratic optimal control problems in terms of structural properties of the control system, such as exponential stabilizability and detectability. The proof relies on the analysis of the exponential convergence of solutions to the differential Riccati equations to the algebraic counterpart, and on a necessary condition for exponential stabilizability in terms of a closed range test.

In this article we propose a compartmental model for the dynamics of Coronavirus Disease 2019 (COVID-19). We take into account the presence of asymptomatic infections and the main policies that have been adopted so far to contain the epidemic: isolation (or social distancing) of a portion of the population, quarantine for confirmed cases and testing. We model isolation by separating the population in two groups: one composed by key-workers that keep working during the pandemic and have a usual contact rate, and a second group consisting of people that are enforced/recommended to stay at home. We refer to quarantine as strict isolation, and it is applied to confirmed infected cases. In the proposed model, the proportion of people in isolation, the level of contact reduction and the testing rate are control parameters that can vary in time, representing policies that evolve in different stages. We obtain an explicit expression for the basic reproduction number $\mathcal{R}_0$ in terms of the parameters of the disease and of the control policies. In this way we can quantify the effect that isolation and testing have in the evolution of the epidemic. We present a series of simulations to illustrate different realistic scenarios. From the expression of $\mathcal{R}_0$ and the simulations we conclude that isolation (social distancing) and testing among asymptomatic cases are fundamental actions to control the epidemic, {and the stricter these measures are and the sooner they are implemented,} the more lives can be saved. Additionally, we show that people that remain in isolation significantly reduce their probability of contagion, so risk groups should be recommended to maintain a low contact rate during the course of the epidemic.