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Mathematical Control and Related Fields

September 2022 , Volume 12 , Issue 3

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Inverse optimal control of regime-switching jump diffusions
Wensheng Yin, Jinde Cao and Yong Ren
2022, 12(3): 567-579 doi: 10.3934/mcrf.2021034 +[Abstract](1153) +[HTML](490) +[PDF](309.19KB)
Abstract:

This paper studies the inverse optimal control using Legendre-Fenchel (in short, LF) translation method for regime-switching jump diffusions. Our approach is to first design inverse pre-optimal stabilization controllers and then obtain inverse optimal stabilizers, which avoids solving a Hamilton-Jacobi-Bellman equation. Finally, an application to stochastic Hamiltonian systems with Markov regime-switching is studied in detail for illustration.

A linear quadratic stochastic Stackelberg differential game with time delay
Weijun Meng and Jingtao Shi
2022, 12(3): 581-609 doi: 10.3934/mcrf.2021035 +[Abstract](1256) +[HTML](575) +[PDF](500.41KB)
Abstract:

This paper is concerned with a linear quadratic stochastic Stackelberg differential game with time delay. The model is general, in which the state delay and the control delay both appear in the state equation, moreover, they both enter into the diffusion term. By introducing two Pseudo-Riccati equations and a special matrix equation, the state feedback representation of the open-loop Stackelberg strategy is derived, under some assumptions. Finally, two examples are given to illustrate the applications of the theoretical results.

Cross-constrained variational method and nonlinear Schrödinger equation with partial confinement
Chenglin Wang and Jian Zhang
2022, 12(3): 611-619 doi: 10.3934/mcrf.2021036 +[Abstract](1104) +[HTML](433) +[PDF](262.6KB)
Abstract:

In this paper, we study the nonlinear Schrödinger equation with a partial confinement. By applying the cross-constrained variational arguments and invariant manifolds of the evolution flow, the sharp condition for global existence and blowup of the solution is derived.

Numerical analysis and simulations of a frictional contact problem with damage and memory
Hailing Xuan and Xiaoliang Cheng
2022, 12(3): 621-639 doi: 10.3934/mcrf.2021037 +[Abstract](1026) +[HTML](430) +[PDF](1720.95KB)
Abstract:

In this paper, we study a frictional contact model which takes into account the damage and the memory. The deformable body consists of a viscoelastic material and the process is assumed to be quasistatic. The mechanical damage of the material which caused by the tension or the compression is included in the constitutive law and the damage function is modelled by a nonlinear parabolic inclusion. Then the variational formulation of the model is governed by a coupled system consisting of a history-dependent hemivariational inequality and a nonlinear parabolic variational inequality. We introduce and study a fully discrete scheme of the problem and derive error estimates for numerical solutions. Under appropriate solution regularity assumptions, an optimal order error estimate is derived for the linear finite element method. Several numerical experiments for the contact problem are given for providing numerical evidence of the theoretical results.

Local null controllability of the penalized Boussinesq system with a reduced number of controls
Jon Asier Bárcena-Petisco and Kévin Le Balc'h
2022, 12(3): 641-666 doi: 10.3934/mcrf.2021038 +[Abstract](1202) +[HTML](411) +[PDF](431.99KB)
Abstract:

In this paper we consider the Boussinesq system with homogeneous Dirichlet boundary conditions, defined in a regular domain \begin{document}$ \Omega\subset\mathbb R^N $\end{document} for \begin{document}$ N = 2 $\end{document} and \begin{document}$ N = 3 $\end{document}. The incompressibility condition of the fluid is replaced by its approximation by penalization with a small parameter \begin{document}$ \varepsilon > 0 $\end{document}. We prove that our system is locally null controllable using a control with a restricted number of components, localized in an open set \begin{document}$ \omega $\end{document} contained in \begin{document}$ \Omega $\end{document}. We also show that the control cost is bounded uniformly with respect to \begin{document}$ \varepsilon \rightarrow 0 $\end{document}. The proof is based on a linearization argument. The null controllability of the linearized system is obtained by proving a new Carleman estimate for the adjoint system. This inequality is derived by exploiting the coercivity of some second order differential operator involving crossed derivatives.

On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback
Julie Valein
2022, 12(3): 667-694 doi: 10.3934/mcrf.2021039 +[Abstract](1028) +[HTML](312) +[PDF](498.23KB)
Abstract:

The aim of this work is to study the asymptotic stability of the nonlinear Korteweg-de Vries equation in the presence of a delayed term in the internal feedback. We first consider the case where the weight of the term with delay is smaller than the weight of the term without delay and we prove a semiglobal stability result for any lengths. Secondly we study the case where the support of the term without delay is not included in the support of the term with delay. In that case, we give a local exponential stability result if the weight of the delayed term is small enough. We illustrate these results by some numerical simulations.

Stable invariant manifolds with application to control problems
Alexey Gorshkov
2022, 12(3): 695-707 doi: 10.3934/mcrf.2021040 +[Abstract](1078) +[HTML](400) +[PDF](308.87KB)
Abstract:

In this article we develop the theory of stable invariant manifolds for evolution equations with application to control problem. We will construct invariant subspaces for linear equations which can be extended to the non-linear equations in the neighbourhood of the equilibrium with help of perturbation theory. Here will be considered both cases of the discrete and continuous spectrum of the generator associated with resolving semi-group. The example of global invariant manifold will be presented for Burgers equation.

A tracking problem for the state of charge in an electrochemical Li-ion battery model
Esteban Hernández, Christophe Prieur and Eduardo Cerpa
2022, 12(3): 709-732 doi: 10.3934/mcrf.2021041 +[Abstract](1223) +[HTML](331) +[PDF](772.57KB)
Abstract:

In this paper the Single Particle Model is used to describe the behavior of a Li-ion battery. The main goal is to design a feedback input current in order to regulate the State of Charge (SOC) to a prescribed reference trajectory. In order to do that, we use the boundary ion concentration as output. First, we measure it directly and then we assume the existence of an appropriate estimator, which has been established in the literature using voltage measurements. By applying backstepping and Lyapunov tools, we are able to build observers and to design output feedback controllers giving a positive answer to the SOC tracking problem. We provide convergence proofs and perform some numerical simulations to illustrate our theoretical results.

Optimal control of transverse vibration of a moving string with time-varying lengths
Bing Sun
2022, 12(3): 733-746 doi: 10.3934/mcrf.2021042 +[Abstract](1152) +[HTML](325) +[PDF](282.84KB)
Abstract:

In this article, we are concerned with optimal control for the transverse vibration of a moving string with time-varying lengths. In the fixed final time horizon case, the Pontryagin maximum principle is established for the investigational system with a moving boundary, owing to the Dubovitskii and Milyutin functional analytical approach. A remark then follows for discussing the utilization of obtained necessary optimality condition.

Convergence of coprime factor perturbations for robust stabilization of Oseen systems
Jan Heiland
2022, 12(3): 747-761 doi: 10.3934/mcrf.2021043 +[Abstract](961) +[HTML](333) +[PDF](488.67KB)
Abstract:

Linearization based controllers for incompressible flows have been proven to work in theory and in simulations. To realize such a controller numerically, the infinite dimensional system has to be linearized and discretized. The unavoidable consistency errors add a small but critical uncertainty to the controller model which will likely make it fail, especially when an observer is involved. Standard robust controller designs can compensate small uncertainties if they can be qualified as a coprime factor perturbation of the plant. We show that for the linearized Navier-Stokes equations, a linearization error can be expressed as a coprime factor perturbation and that this perturbation smoothly depends on the size of the linearization error. In particular, improving the linearization makes the perturbation smaller so that, eventually, standard robust controller will stabilize the system.

Asymptotic gain results for attractors of semilinear systems
Jochen Schmid, Oleksiy Kapustyan and Sergey Dashkovskiy
2022, 12(3): 763-788 doi: 10.3934/mcrf.2021044 +[Abstract](932) +[HTML](387) +[PDF](434.48KB)
Abstract:

We establish asymptotic gain along with input-to-state practical stability results for disturbed semilinear systems w.r.t. the global attractor of the respective undisturbed system. We apply our results to a large class of nonlinear reaction-diffusion equations comprising disturbed Chaffee–Infante equations, for example.

A nonlinear version of Halanay's inequality for the uniform convergence to the origin
Pierdomenico Pepe
2022, 12(3): 789-811 doi: 10.3934/mcrf.2021045 +[Abstract](1285) +[HTML](456) +[PDF](370.99KB)
Abstract:

A nonlinear version of Halanay's inequality is studied in this paper as a sufficient condition for the convergence of functions to the origin, uniformly with respect to bounded sets of initial values. The same result is provided in the case of forcing terms, for the uniform convergence to suitable neighborhoods of the origin. Related Lyapunov methods for the global uniform asymptotic stability and the input-to-state stability of systems described by retarded functional differential equations, with possibly nonconstant time delays, are provided. The relationship with the Razumikhin methodology is shown.

Computation of open-loop inputs for uniformly ensemble controllable systems
Michael Schönlein
2022, 12(3): 813-829 doi: 10.3934/mcrf.2021046 +[Abstract](910) +[HTML](304) +[PDF](343.35KB)
Abstract:

This paper presents computational methods for families of linear systems depending on a parameter. Such a family is called ensemble controllable if for any family of parameter-dependent target states and any neighborhood of it there is a parameter-independent input steering the origin into the neighborhood. Assuming that a family of systems is ensemble controllable we present methods to construct suitable open-loop input functions. Our approach to solve this infinite-dimensional task is based on a combination of methods from the theory of linear integral equations and finite-dimensional control theory.

Controllability to rest of the Gurtin-Pipkin model
Xiuxiang Zhou and Shu Luan
2022, 12(3): 831-845 doi: 10.3934/mcrf.2021051 +[Abstract](746) +[HTML](268) +[PDF](285.59KB)
Abstract:

This paper is devoted to analyzing the controllability to rest of the Gurtin-Pipkin model, which is a class of differential equations with memory terms. The goal is not only to derive the state to vanish at some time but also to require the memory term to vanish at the same time, ensuring that the controlled system is controllable to rest. In order to get rid of the influence of memory, the controllability result is obtained by means of the Fourier type approach and the moment theory.

2021 Impact Factor: 1.141
5 Year Impact Factor: 1.362
2021 CiteScore: 2.4

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