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Optimal control problems without terminal constraints: The turnpike property with interior decay

Gugat Martin, Lazar Martin

International Journal of Applied Mathematics and Computer Science

2023

Vol. 33, no. 3

429--438

We show a turnpike result for problems of optimal control with possibly nonlinear systems as well as pointwise-in-time state and control constraints. The objective functional is of integral type and contains a tracking term which penalizes the distance to a desired steady state. In the optimal control problem, only the initial state is prescribed. We assume that a cheap control condition holds that yields a bound for the optimal value of our optimal control problem in terms of the initial data. We show that the solutions to the optimal control problems on the time intervals [0, T] have a turnpike structure in the following sense: For large T the contribution to the objective functional that comes from the subinterval [T/2, T], i.e., from the second half of the time interval [0, T], is at most of the order 1/T. More generally, the result holds for subintervals of the form [r T,T], where r ∈ (0, 1/2) is a real number. Using this result inductively implies that the decay of the integral on such a subinterval in the objective function is faster than the reciprocal value of a power series in T with positive coefficients. Accordingly, the contribution to the objective value from the final part of the time interval decays rapidly with a growing time horizon. At the end of the paper we present examples for optimal control problems where our results are applicable.

Riesz basis of exponential family for a hyperbolic system

Intissar Abdelkader, Jeribi Aref, Walha Ines

Journal of Applied Analysis

2019

Vol. 25, nr 1

13--23

This paper studies a linear hyperbolic system with boundary conditions thatwas first studied under someweaker conditions in [8, 11]. Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. It is shown that the associated linear system is the infinitesimal generator of a C0-semigroup; its spectrum consists of zeros of a sine-type function, and its exponential system {eλnt}n≥1 constitutes a Riesz basis in L2[0, T]. Furthermore, by the spectral analysis method, it is also shown that the linear system has a sequence of eigenvectors, which form a Riesz basis in Hilbert space, and hence the spectrum-determined growth condition is deduced.

Transmitancje operatorowe pewnej klasy układów typu hiperbolicznego z wejściami brzegowymi

Bartecki K.

Pomiary Automatyka Kontrola

2014

R. 60, nr 1

49--52

W artykule przedstawiono ogólną postać transmitancji operatorowych pewnej klasy układów o parametrach rozłożonych, opisanych dwoma równaniami różniczkowymi cząstkowymi typu hiperbolicznego. Zakładając istnienie w układzie dwóch wejść o charakterze wymuszeń brzegowych typu Dirichleta oraz dwóch wyjść rozłożonych wzdłuż osi zmiennej przestrzennej, przedstawiono wyrażenia opisujące transmitancje operatorowe układu dla dwóch różnych konfiguracji sygnałów wejściowych. Rozważania zilustrowano praktycznym przykładem wymiennika ciepła pracującego w układach: współ- oraz przeciwprądowym.

Transfer function models for a class of distributed parameter systems described by the two hyperbolic partial differential equations defined on a one-dimensional finite spatial domain are considered. Assuming two boundary inputs of Dirichlet type, the closed-form expressions for the individual elements of the 22 transfer function matrix are proposed based on the decoupled canonical representation of the system. The influence of the location of the boundary inputs on the transfer function representation is demonstrated for two different input configurations. The first one is the so-called congruent arrangement, for which both inputs act on the system at the same spatial position, l=0 (Fig. 1). The second one is the incongruent arrangement, where both inputs act on the system at its opposite ends, l=0 and l=L, respectively (Fig. 2). The considerations are illustrated with a practical example of a shell and tube heat exchanger operating in parallel- and countercurrent-flow modes (Fig. 3), which correspond to the two abovementioned boundary input configurations. Based on the transfer function model, both frequency and time responses of the system can be determined, which can be useful e.g. in the case of the model-based fault detection scheme.