# American Institute of Mathematical Sciences

eISSN:
2163-2480

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## Evolution Equations and Control Theory

August 2022 , Volume 11 , Issue 4

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2022, 11(4): 1001-1035 doi: 10.3934/eect.2021034 +[Abstract](1300) +[HTML](485) +[PDF](453.43KB)
Abstract:

The present paper proposes, in a real separable Hilbert space, to analyze the existence of solutions for a class of perturbed second-order state-dependent maximal monotone operators with a finite delay. The dependence of the operators is -in some sense- absolutely continuous (or bounded continuous) variation in time, and Lipschitz continuous in state. The approach to solve our problem is based on a discretization scheme. The uniqueness result is applied to optimal control.

2022, 11(4): 1037-1069 doi: 10.3934/eect.2021035 +[Abstract](1206) +[HTML](705) +[PDF](473.79KB)
Abstract:

In this paper, we deal with fractional neutral evolution systems of hyperbolic type in Banach spaces. We establish the existence and uniqueness of the mild solution and prove the approximate controllability of the systems under different conditions. These results are mainly based on fixed point theorems as well as constructing a Cauchy sequence and a control function. In the end, we give an example to illustrate the validity of the main results.

2022, 11(4): 1071-1086 doi: 10.3934/eect.2021036 +[Abstract](1061) +[HTML](428) +[PDF](335.7KB)
Abstract:

This paper studies a coupled system of plate equations with variable coefficients, subject to the clamped boundary conditions. By the Riemannian geometry approach, the duality method, the multiplier technique and a compact perturbation method, we establish exact boundary null controllability of the system under verifiable assumptions.

2022, 11(4): 1087-1148 doi: 10.3934/eect.2021037 +[Abstract](1133) +[HTML](515) +[PDF](823.66KB)
Abstract:

Backstepping based controller and observer models were designed for higher order linear and nonlinear Schrödinger equations on a finite interval in [3] where the controller was assumed to be acting from the left endpoint of the medium. In this companion paper, we further the analysis by considering boundary controller(s) acting at the right endpoint of the domain. It turns out that the problem is more challenging in this scenario as the associated boundary value problem for the backstepping kernel becomes overdetermined and lacks a smooth solution. The latter is essential to switch back and forth between the original plant and the so called target system. To overcome this difficulty we rely on the strategy of using an imperfect kernel, namely one of the boundary conditions in kernel PDE model is disregarded. The drawback is that one loses rapid stabilization in comparison with the left endpoint controllability. Nevertheless, the exponential decay of the \begin{document}$L^2$\end{document}-norm with a certain rate still holds. The observer design is associated with new challenges from the point of view of wellposedness and one has to prove smoothing properties for an associated initial boundary value problem with inhomogeneous boundary data. This problem is solved by using Laplace transform in time. However, the Bromwich integral that inverts the transformed solution is associated with certain analyticity issues which are treated through a subtle analysis. Numerical algorithms and simulations verifying the theoretical results are given.

2022, 11(4): 1149-1173 doi: 10.3934/eect.2021038 +[Abstract](1093) +[HTML](577) +[PDF](448.61KB)
Abstract:

In this paper, we consider a Balakrishnan-Taylor viscoelastic wave equation with nonlinear frictional damping and logarithmic source term. By assuming a more general type of relaxation functions, we establish explicit and general decay rate results, using the multiplier method and some properties of the convex functions. This result is new and generalizes earlier results in the literature.

2022, 11(4): 1175-1190 doi: 10.3934/eect.2021039 +[Abstract](882) +[HTML](383) +[PDF](473.06KB)
Abstract:

The present work is based on the coupling of prion proliferation system together with chaperone which consists of two ODEs and a partial integro-differential equation. The existence and uniqueness of a positive global classical solution of the system is proved for the bounded degradation rates by the idea of evolution system theory in the state space \begin{document}$\mathbb{R} \times \mathbb{R} \times L_{1}(Z,zdz).$\end{document} Moreover, the global weak solutions for unbounded degradation rates are discussed by weak compactness technique.

2022, 11(4): 1191-1200 doi: 10.3934/eect.2021040 +[Abstract](1176) +[HTML](393) +[PDF](324.43KB)
Abstract:

Stochastic functional differential equations with infinite delay are considered. A novel approach to exponential stability of such equations is proposed. New criteria for the mean square exponential stability of general stochastic functional differential equations with infinite delay are presented. Illustrative examples are given.

2022, 11(4): 1201-1227 doi: 10.3934/eect.2021041 +[Abstract](1075) +[HTML](366) +[PDF](360.73KB)
Abstract:

In this paper, we investigate the optimal time-decay rates of global classical solutions for the compressible Oldroyd-B model in \begin{document}$\mathbb{R}^n(n = 2,3)$\end{document}. Global classical solutions in two space dimensions are still open. We first complete the proof of global classical solutions in two space dimensions. Based on global classical solutions and Fourier spectrum analysis, we obtain the optimal time-decay rates of global classical solutions in two and three space dimensions. More precisely, if the initial data belong to \begin{document}$L^1$\end{document}, the optimal time-decay rate of solutions and time-decay rates of \begin{document}$l(l = 1,\cdot\cdot\cdot,m)$\end{document} order derivatives under additional assumptions are established.

2022, 11(4): 1229-1249 doi: 10.3934/eect.2021042 +[Abstract](878) +[HTML](374) +[PDF](416.52KB)
Abstract:

The paper deals with quadratic optimal control problems, we study the equivalence between well-posed problems and affinity on the control for a second-order differential inclusions with two-points conditions, governed by a maximal monotone operator in a finite dimensional space.

2022, 11(4): 1251-1283 doi: 10.3934/eect.2021043 +[Abstract](1074) +[HTML](470) +[PDF](3453.37KB)
Abstract:

We investigate the null controllability property of systems that mathematically describe the dynamics of some non-Newtonian incompressible viscous flows. The principal model we study was proposed by O. A. Ladyzhenskaya, although the techniques we develop here apply to other fluids having a shear-dependent viscosity. Taking advantage of the Pontryagin Minimum Principle, we utilize a bootstrapping argument to prove that sufficiently smooth controls to the forced linearized Stokes problem exist, as long as the initial data in turn has enough regularity. From there, we extend the result to the nonlinear problem. As a byproduct, we devise a quasi-Newton algorithm to compute the states and a control, which we prove to converge in an appropriate sense. We finish the work with some numerical experiments.

2022, 11(4): 1285-1307 doi: 10.3934/eect.2021044 +[Abstract](975) +[HTML](416) +[PDF](469.01KB)
Abstract:

This paper deals with the hierarchical control of the anisotropic heat equation with dynamic boundary conditions and drift terms. We use the Stackelberg-Nash strategy with one leader and two followers. To each fixed leader, we find a Nash equilibrium corresponding to a bi-objective optimal control problem for the followers. Then, by some new Carleman estimates, we prove a null controllability result.

2022, 11(4): 1309-1329 doi: 10.3934/eect.2021045 +[Abstract](1004) +[HTML](405) +[PDF](406.51KB)
Abstract:

In this paper, we introduce and study a class of delay differential variational inequalities comprising delay differential equations and variational inequalities. We establish a sufficient condition for the existence of periodic solutions to delay differential variational inequalities. Based on some fixed point arguments, in both single-valued and multivalued cases, the solvability of initial value and periodic problems are proved. Furthermore, we study the conditional stability of periodic solutions to this systems.

2022, 11(4): 1331-1355 doi: 10.3934/eect.2021046 +[Abstract](1000) +[HTML](314) +[PDF](462.42KB)
Abstract:

The paper is concerned with a system of linear hyperbolic differential equations on a network coupled through general transmission conditions of Kirchhoff's-type at the nodes. We discuss the reduction of such a problem to a system of 1-dimensional hyperbolic problems for the associated Riemann invariants and provide a semigroup-theoretic proof of its well-posedness. A number of examples showing the relation of our results with recent research is also provided.

2022, 11(4): 1357-1372 doi: 10.3934/eect.2021047 +[Abstract](909) +[HTML](413) +[PDF](390.76KB)
Abstract:

This paper deals with stability of solution map to a parametric control problem governed by semilinear elliptic equations with finite unilateral constraints, where the objective functional is not convex. By using the first-order necessary optimality conditions, we derive some sufficient conditions under which the solution map is upper semicontinuous with respect to parameters.

2022, 11(4): 1373-1398 doi: 10.3934/eect.2021048 +[Abstract](922) +[HTML](346) +[PDF](385.44KB)
Abstract:

The problems of designing feedback control algorithms for parabolic and hyperbolic variational inequalities are considered. These algorithms should preserve given properties of solutions of inequalities under the action of unknown disturbances. Solving algorithms that are stable with respect to informational noises are constructed. The algorithms are based on the method of extremal shift well-known in the theory of guaranteed control.

2022, 11(4): 1399-1454 doi: 10.3934/eect.2021049 +[Abstract](1625) +[HTML](288) +[PDF](581.53KB)
Abstract:

We consider a nonlinear evolution equation in the form

together with its singular limit problem as \begin{document}$\varepsilon\to 0$\end{document}

where \begin{document}$\varepsilon\in (0,1]$\end{document} (possibly \begin{document}$\varepsilon = 0$\end{document}), \begin{document}${\rm A}_\varepsilon$\end{document} and \begin{document}${\rm A}$\end{document} are linear closed (possibly) unbounded operators, \begin{document}${\rm N}_\varepsilon$\end{document} and \begin{document}${\rm N}$\end{document} are linear (possibly) unbounded operators, \begin{document}${\rm G}_\varepsilon$\end{document} and \begin{document}${\rm G}$\end{document} are nonlinear functions. We give sufficient conditions on \begin{document}${\rm A}_\varepsilon,$\end{document} \begin{document}${\rm N}_\varepsilon$\end{document} and \begin{document}${\rm G}_\varepsilon$\end{document} (and also \begin{document}${\rm A}$\end{document}, \begin{document}${\rm N}$\end{document} and \begin{document}${\rm G}$\end{document}) that guarantee not only the existence of an inertial manifold of dimension independent of \begin{document}$\varepsilon$\end{document} for (Eε) on a Banach space \begin{document}${\mathcal H}$\end{document}, but also the Hölder continuity, lower and upper semicontinuity at \begin{document}$\varepsilon = 0$\end{document} of the intersection of the inertial manifold with a bounded absorbing set. Applications to higher order viscous Cahn-Hilliard-Oono equations, the hyperbolic type equations and the phase-field systems, subject to either Neumann or Dirichlet boundary conditions (BC) (in which case \begin{document}$\Omega\subset{\mathbb R}^d$\end{document} is a bounded domain with smooth boundary) or periodic BC (in which case \begin{document}$\Omega = \Pi_{i = 1}^d(0,L_i),$\end{document} \begin{document}$L_i>0$\end{document}), \begin{document}$d = 1,2\; {\rm or} \;3$\end{document}, are considered. These three classes of dissipative equations read

and

respectively, where \begin{document}$\sigma\ge 0$\end{document} and the Laplace operator is defined as

We assume that, for a given real number \begin{document}${\frak c}_1>0,$\end{document} there exists a positive integer \begin{document}$n = n({\frak c}_1)$\end{document} such that \begin{document}$\lambda_{n+1}-\lambda_n>{\frak c}_1$\end{document}, where \begin{document}$\{\lambda_k\}_{k\in\mathbb N^*}$\end{document} are the eigenvalues of \begin{document}$N$\end{document}. There exists a real number \begin{document}${\mathscr M}>0$\end{document} such that the nonlinear function \begin{document}$g: V_j\to V_j$\end{document} satisfies the conditions \begin{document}$\|g(\psi)\|_j\le {\mathscr M}$\end{document} and \begin{document}$\|g(\psi)-g(\varphi)\|_{V_j}\le {\mathscr M}\|\psi-\varphi\|_{V_j}$\end{document}, \begin{document}$\forall\psi,\varphi\in V_j$\end{document}, where \begin{document}$V_j = {\mathscr D}(N^{j/2})$\end{document}, \begin{document}$j = 1$\end{document} for Problems (Pε) and (Sε) and \begin{document}$j = 0, 2\alpha$\end{document} for Problem (Hε). We further require \begin{document}$g\in{\mathcal C}^1( V_j, V_j)$\end{document}, \begin{document}$\|g'(\psi)\varphi\|_j\le {\mathscr M}\|\varphi\|_j$\end{document} for Problems (Hε) and (Sε).

2022, 11(4): 1455-1487 doi: 10.3934/eect.2021050 +[Abstract](996) +[HTML](269) +[PDF](451.48KB)
Abstract:

For \begin{document}$\nu\in(0,1)$\end{document}, we investigate the nonautonomous subdiffusion equation:

where \begin{document}$\mathbf{D}_{t}^{\nu}$\end{document} is the Caputo fractional derivative and \begin{document}$\mathcal{L}$\end{document} is a uniformly elliptic operator with smooth coefficients depending on time. Under suitable conditions on the given data and a minimal number (i.e. the necessary number) of compatibility conditions, the global classical solvability to the related initial-boundary value problems are established in the weighted fractional Hölder spaces.

2021 Impact Factor: 1.169
5 Year Impact Factor: 1.294
2021 CiteScore: 2