# American Institute of Mathematical Sciences

ISSN:
2156-8472

eISSN:
2156-8499

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## Mathematical Control & Related Fields

June 2018 , Volume 8 , Issue 2

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2018, 8(2): 383-395 doi: 10.3934/mcrf.2018015 +[Abstract](476) +[HTML](274) +[PDF](434.73KB)
Abstract:

Design of controller subject to a constraint for a Schrödinger equation is considered based on the energy functional of the system. Thus, the resulting closed-loop system is nonlinear and its well-posedness is proven by the nonlinear monotone operator theory and a complex form of the nonlinear Lax-Milgram theorem. The asymptotic stability and exponential stability of the system are discussed with the LaSalle invariance principle and Riesz basis method, respectively. In the end, a numerical simulation illustrates the feasibility of the suggested feedback control law.

2018, 8(2): 397-410 doi: 10.3934/mcrf.2018016 +[Abstract](400) +[HTML](225) +[PDF](410.77KB)
Abstract:

In this article we study the controllability properties of general compactly perturbed exactly controlled linear systems with admissible control operators. Firstly, we show that approximate and exact controllability are equivalent properties for such systems. Then, and more importantly, we provide for the perturbed system a complete characterization of the set of reachable states in terms of the Fattorini-Hautus test. The results rely on the Peetre lemma.

2018, 8(2): 411-449 doi: 10.3934/mcrf.2018017 +[Abstract](459) +[HTML](223) +[PDF](1072.35KB)
Abstract:

This work is concerned with the optimal control problems governed by a 1D wave equation with variable coefficients and the control spaces \begin{document}$\mathcal M_T$\end{document} of either measure-valued functions \begin{document}$L_{{{w}^{*}}}^{2}\left( I, \mathcal M\left( {\mathit \Omega } \right) \right)$\end{document} or vector measures \begin{document}$\mathcal M({\mathit \Omega }, L^2(I))$\end{document}. The cost functional involves the standard quadratic tracking terms and the regularization term \begin{document}$α\|u\|_{\mathcal M_T}$\end{document} with \begin{document}$α>0$\end{document}. We construct and study three-level in time bilinear finite element discretizations for this class of problems. The main focus lies on the derivation of error estimates for the optimal state variable and the error measured in the cost functional. The analysis is mainly based on some previous results of the authors. The numerical results are included.

2018, 8(2): 451-473 doi: 10.3934/mcrf.2018018 +[Abstract](417) +[HTML](249) +[PDF](436.85KB)
Abstract:

In this paper, we study the generalized mean-field stochastic control problem when the usual stochastic maximum principle (SMP) is not applicable due to the singularity of the Hamiltonian function. In this case, we derive a second order SMP. We introduce the adjoint process by the generalized mean-field backward stochastic differential equation. The keys in the proofs are the expansion of the cost functional in terms of a perturbation parameter, and the use of the range theorem for vector-valued measures.

2018, 8(2): 475-490 doi: 10.3934/mcrf.2018019 +[Abstract](416) +[HTML](217) +[PDF](253.58KB)
Abstract:

This paper is concerned with the admissibility analysis and control synthesis for a class of singular systems with Markovian jumps and time-varying delay. The basic idea is the use of an augmented Lyapunov-Krasovskii functional together with a series of appropriate integral inequalities. Sufficient conditions are established to ensure the systems to be admissible. Moreover, control design via static output feedback (SOF) is derived to achieve the stabilization for singular systems. A new algorithm is built to solve the SOF controllers. Examples are given to show the effectiveness of the proposed method.

2017  Impact Factor: 0.631