# American Institute of Mathematical Sciences

ISSN:
2156-8472

eISSN:
2156-8499

All Issues

## Mathematical Control and Related Fields

March 2022 , Volume 12 , Issue 1

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2022, 12(1): 1-16 doi: 10.3934/mcrf.2020055 +[Abstract](1348) +[HTML](441) +[PDF](590.71KB)
Abstract:

This paper deals with active noise control applied to a one-dimensional acoustic propagation system. The aim here is to keep over time a zero noise level at a given point. We aim to design this control using noise measurement at some point in the spatial domain. Based on symmetry property, we are able to design a feedback boundary control allowing this fact. Moreover, using D'Alembert formula, an explicit formula of the control can be computed. Even if the focus is made on the wave equation, this approach is easily extendable to more general operators.

2022, 12(1): 17-47 doi: 10.3934/mcrf.2021001 +[Abstract](1149) +[HTML](411) +[PDF](732.01KB)
Abstract:

A low-gain integral controller with anti-windup component is presented for exponentially stable, linear, discrete-time, infinite-dimensional control systems subject to input nonlinearities and external disturbances. We derive a disturbance-to-state stability result which, in particular, guarantees that the tracking error converges to zero in the absence of disturbances. The discrete-time result is then used in the context of sampled-data low-gain integral control of stable well-posed linear infinite-dimensional systems with input nonlinearities. The sampled-date control scheme is applied to two examples (including sampled-data control of a heat equation on a square) which are discussed in some detail.

2022, 12(1): 49-80 doi: 10.3934/mcrf.2021002 +[Abstract](1265) +[HTML](425) +[PDF](804.95KB)
Abstract:

In this work, we are interested in tree-shaped networks of freely vibrating beams which are geometrically exact (GEB) – in the sense that large motions (deflections, rotations) are accounted for in addition to shearing – and linked by rigid joints. For the intrinsic GEB formulation, namely that in terms of velocities and internal forces/moments, we derive transmission conditions and show that the network is locally in time well-posed in the classical sense. Applying velocity feedback controls at the external nodes of a star-shaped network, we show by means of a quadratic Lyapunov functional and the theory developed by Bastin & Coron in [2] that the zero steady state of this network is exponentially stable for the \begin{document}$H^1$\end{document} and \begin{document}$H^2$\end{document} norms. The major obstacles to overcome in the intrinsic formulation of the GEB network, are that the governing equations are semilinar, containing a quadratic nonlinearity, and that linear lower order terms cannot be neglected.

2022, 12(1): 81-114 doi: 10.3934/mcrf.2021003 +[Abstract](1467) +[HTML](457) +[PDF](535.85KB)
Abstract:

A learning approach for determining which operator from a class of nonlocal operators is optimal for the regularization of an inverse problem is investigated. The considered class of nonlocal operators is motivated by the use of squared fractional order Sobolev seminorms as regularization operators. First fundamental results from the theory of regularization with local operators are extended to the nonlocal case. Then a framework based on a bilevel optimization strategy is developed which allows to choose nonlocal regularization operators from a given class which i) are optimal with respect to a suitable performance measure on a training set, and ii) enjoy particularly favorable properties. Results from numerical experiments are also provided.

2022, 12(1): 115-146 doi: 10.3934/mcrf.2021004 +[Abstract](1283) +[HTML](365) +[PDF](1394.92KB)
Abstract:

We propose a novel numerical algorithm utilizing model reduction for computing solutions to stationary partial differential equations involving the spectral fractional Laplacian. Our approach utilizes a known characterization of the solution in terms of an integral of solutions to local (classical) elliptic problems. We reformulate this integral into an expression whose continuous and discrete formulations are stable; the discrete formulations are stable independent of all discretization parameters. We subsequently apply the reduced basis method to accomplish model order reduction for the integrand. Our choice of quadrature in discretization of the integral is a global Gaussian quadrature rule that we observe is more efficient than previously proposed quadrature rules. Finally, the model reduction approach enables one to compute solutions to multi-query fractional Laplace problems with orders of magnitude less cost than a traditional solver.

2022, 12(1): 147-168 doi: 10.3934/mcrf.2021005 +[Abstract](1181) +[HTML](361) +[PDF](386.3KB)
Abstract:

We study the stability of reaction-diffusion equations in presence of noise. The relationship of stability of solutions between the stochastic ordinary different equations and the corresponding stochastic reaction-diffusion equation is firstly established. Then, by using the Lyapunov method, sufficient conditions for mean square and stochastic stability are given. The results show that the multiplicative noise can make the solution stable, but the additive noise will be not.

2022, 12(1): 169-200 doi: 10.3934/mcrf.2021006 +[Abstract](1137) +[HTML](354) +[PDF](591.7KB)
Abstract:

We use a variant the backstepping method to study the stabilization of a 1-D linear transport equation on the interval \begin{document}$(0,L)$\end{document}, by controlling the scalar amplitude of a piecewise regular function of the space variable in the source term. We prove that if the system is controllable in a periodic Sobolev space of order greater than \begin{document}$1$\end{document}, then the system can be stabilized exponentially in that space and, for any given decay rate, we give an explicit feedback law that achieves that decay rate. The variant of the backstepping method used here relies mainly on the spectral properties of the linear transport equation, and leads to some original technical developments that differ substantially from previous applications.

2022, 12(1): 201-223 doi: 10.3934/mcrf.2021007 +[Abstract](1303) +[HTML](386) +[PDF](483.88KB)
Abstract:

We introduce a general system of ordinary differential equations that includes some classical and recent models for the epidemic spread in a closed population without vital dynamic in a finite time horizon. The model is vectorial, in the sense that it accounts for a vector valued state function whose components represent various kinds of exposed/infected subpopulations, with a corresponding vector of control functions possibly different for any subpopulation. In the general setting, we prove well-posedness and positivity of the initial value problem for the system of state equations and the existence of solutions to the optimal control problem of the coefficients of the nonlinear part of the system, under a very general cost functional. We also prove the uniqueness of the optimal solution for a small time horizon when the cost is superlinear in all control variables with possibly different exponents in the interval \begin{document}$(1,2]$\end{document}. We consider then a linear cost in the control variables and study the singular arcs. Full details are given in the case \begin{document}$n = 1$\end{document} and the results are illustrated by the aid of some numerical simulations.

2022, 12(1): 225-243 doi: 10.3934/mcrf.2021019 +[Abstract](996) +[HTML](374) +[PDF](363.74KB)
Abstract:

In the paper, a new Gronwall lemma for functions of two variables with singular integrals is proved. An application to weak relative compactness of the set of solutions to a fractional partial differential equation is given.

2022, 12(1): 245-273 doi: 10.3934/mcrf.2021021 +[Abstract](1117) +[HTML](354) +[PDF](630.24KB)
Abstract:

This paper addresses the following question: "Suppose that a state-feedback controller stabilizes an infinite-dimensional linear continuous-time system. If we choose the parameters of an event/self-triggering mechanism appropriately, is the event/self-triggered control system stable under all sufficiently small nonlinear Lipschitz perturbations?" We assume that the stabilizing feedback operator is compact. This assumption is used to guarantee the strict positiveness of inter-event times and the existence of the mild solution of evolution equations with unbounded control operators. First, for the case where the control operator is bounded, we show that the answer to the above question is positive, giving a sufficient condition for exponential stability, which can be employed for the design of event/self-triggering mechanisms. Next, we investigate the case where the control operator is unbounded and prove that the answer is still positive for periodic event-triggering mechanisms.

2020 Impact Factor: 1.284
5 Year Impact Factor: 1.345
2020 CiteScore: 1.9