# American Institute of Mathematical Sciences

ISSN:
2156-8472

eISSN:
2156-8499

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## Mathematical Control & Related Fields

June 2020 , Volume 10 , Issue 2

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2020, 10(2): 217-256 doi: 10.3934/mcrf.2019037 +[Abstract](1384) +[HTML](484) +[PDF](1864.5KB)
Abstract:

The main goal of this paper is to investigate the controllability properties of semi-discrete in space coupled parabolic systems with less controls than equations, in dimension greater than \begin{document}$1$\end{document}. We are particularly interested in the boundary control case which is notably more intricate that the distributed control case, even though our analysis is more general.

The main assumption we make on the geometry and on the evolution equation itself is that it can be put into a tensorized form. In such a case, following [5] and using an adapted version of the Lebeau-Robbiano construction, we are able to prove controllability results for those semi-discrete systems (provided that the structure of the coupling terms satisfies some necessary Kalman condition) with uniform bounds on the controls.

To achieve this objective we actually propose an abstract result on ordinary differential equations with estimates on the control and the solution whose dependence upon the system parameters are carefully tracked. When applied to an ODE coming from the discretization in space of a parabolic system, we thus obtain uniform estimates with respect to the discretization parameters.

Yuan Wu and
2020, 10(2): 257-274 doi: 10.3934/mcrf.2019038 +[Abstract](1325) +[HTML](536) +[PDF](1276.41KB)
Abstract:

In this paper, under the structure framework, a valuation model for a corporate bond with credit rating migration risk and in macro regime switch is established. The model turns to a free boundary problem in a partial differential equation (PDE) system. By PDE techniques, the existence, uniqueness and regularity of the solution are obtained. Furthermore, numerical examples are also presented.

2020, 10(2): 275-304 doi: 10.3934/mcrf.2019039 +[Abstract](1422) +[HTML](585) +[PDF](536.72KB)
Abstract:

In this work we consider the exact controllability and the stabilization for the generalized Benney-Luke equation

on a periodic domain \begin{document}$S$\end{document} (the unit circle on the plane) with internal control \begin{document}$f$\end{document} supported on an arbitrary sub-domain of \begin{document}$S$\end{document}. We establish that the model is exactly controllable in a Sobolev type space when the whole \begin{document}$S$\end{document} is the support of \begin{document}$f$\end{document}, without any assumption on the size of the initial and final states, and that the model is local exactly controllable when the support of \begin{document}$f$\end{document} is a proper subdomain of \begin{document}$S$\end{document}, assuming that initial and terminal states are small. Moreover, in the case that the initial data is small and \begin{document}$f$\end{document} is a special internal linear feedback, the solution of the model must have uniform exponential decay to a constant state.

2020, 10(2): 305-331 doi: 10.3934/mcrf.2019040 +[Abstract](1618) +[HTML](510) +[PDF](537.78KB)
Abstract:

We consider a particular phase field system which physical context is that of tumor growth dynamics. The model we deal with consists of a Cahn-Hilliard equation governing the evolution of the phase variable which takes into account the tumor cells proliferation in the tissue coupled with a reaction-diffusion equation for the nutrient. This model has already been investigated from the viewpoint of well-posedness, long-time behavior, and asymptotic analyses as some parameters go to zero. Starting from these results, we aim to face a related optimal control problem by employing suitable asymptotic schemes. In this direction, we assume some quite general growth conditions for the involved potential and a smallness restriction for a parameter appearing in the system we are going to face. We provide the existence of optimal controls and a necessary condition for optimality is addressed.

2020, 10(2): 333-363 doi: 10.3934/mcrf.2019041 +[Abstract](1586) +[HTML](544) +[PDF](677.34KB)
Abstract:

We consider an optimal control problem governed by a one-dimensional elliptic equation that involves univariate functions of bounded variation as controls. For the discretization of the state equation we use linear finite elements and for the control discretization we analyze two strategies. First, we use variational discretization of the control and show that the \begin{document}$L^2$\end{document}- and \begin{document}$L^\infty$\end{document}-error for the state and the adjoint state are of order \begin{document}${\mathcal O}(h^2)$\end{document} and that the \begin{document}$L^1$\end{document}-error of the control behaves like \begin{document}${\mathcal O}(h^2)$\end{document}, too. These results rely on a structural assumption that implies that the optimal control of the original problem is piecewise constant and that the adjoint state has nonvanishing first derivative at the jump points of the control. If, second, piecewise constant control discretization is used, we obtain \begin{document}$L^2$\end{document}-error estimates of order \begin{document}$\mathcal{O}(h)$\end{document} for the state and \begin{document}$W^{1, \infty}$\end{document}-error estimates of order \begin{document}$\mathcal{O}(h)$\end{document} for the adjoint state. Under the same structural assumption as before we derive an \begin{document}$L^1$\end{document}-error estimate of order \begin{document}$\mathcal{O}(h)$\end{document} for the control. We discuss optimization algorithms and provide numerical results for both discretization schemes indicating that the error estimates are optimal.

2020, 10(2): 365-378 doi: 10.3934/mcrf.2020001 +[Abstract](1441) +[HTML](346) +[PDF](444.9KB)
Abstract:

In this paper, we investigate the optimal investment problem in the presence of delay under partial information. We assume that the financial market consists of one risk free asset (bond) and one risky asset (stock) and only the price of the risky asset can be observed from the financial market. The objective of the investor is to maximize the expected utility of the terminal wealth and average of the path segment. By using the filtering theory, we establish the separation principle and reduce the problem to the complete information case. Explicit expressions for the value function and the corresponding optimal strategy are obtained by solving the corresponding Hamilton-Jacobi-Bellman equation. Furthermore, we study the sensitivity of the optimal investment strategy on the model parameters in a numerical section and both of the full and partial information schemes are simulated and compared.

2020, 10(2): 379-403 doi: 10.3934/mcrf.2020002 +[Abstract](1154) +[HTML](384) +[PDF](471.96KB)
Abstract:

The aim of this paper is to establish a necessary condition for optimal stochastic controls where the systems governed by forward-backward doubly stochastic differential equations (FBDSDEs in short). The control constraints need not to be convex. This condition is described by two kinds of new adjoint processes containing two Brownian motions, corresponding to the forward and backward components and a maximum condition on the Hamiltonian. The proof of the main result is based on spike's variational principle, duality technique and delicate estimates on the state and the adjoint processes with respect to the control variable. An example is provided for illustration.

2020, 10(2): 405-424 doi: 10.3934/mcrf.2020003 +[Abstract](1260) +[HTML](359) +[PDF](980.74KB)
Abstract:

We consider an interlinked production model consisting of conservation laws (PDE) coupled to ordinary differential equations (ODE). Our focus is the analysis of control laws for the coupled system and corresponding stabilization questions of equilibrium dynamics in the presence of disturbances. These investigations are carried out using an appropriate Lyapunov function on the theoretical and numerical level. The discrete \begin{document}$L^2-$\end{document}stabilization technique allows to derive a mixed feedback law that is able to ensure exponential stability also in bottleneck situations. All results are accompanied by computational examples.

2020, 10(2): 425-441 doi: 10.3934/mcrf.2020004 +[Abstract](976) +[HTML](339) +[PDF](395.41KB)
Abstract:

We consider the problem of optimal singular control of a stochastic partial differential equation (SPDE) with space-mean dependence. Such systems are proposed as models for population growth in a random environment. We obtain sufficient and necessary maximum principles for these control problems. The corresponding adjoint equation is a reflected backward stochastic partial differential equation (BSPDE) with space-mean dependence. We prove existence and uniqueness results for such equations. As an application we study optimal harvesting from a population modelled as an SPDE with space-mean dependence.

2020, 10(2): 443-454 doi: 10.3934/mcrf.2020005 +[Abstract](881) +[HTML](345) +[PDF](355.55KB)
Abstract:

We analyze the exact controllability problem of switching controls for stochastic control systems endowed with different actuators. The goal is to control the dynamics of the system by switching from an actuator to the other such that, in each instant of time, there are as few active actuators as possible. We prove that, under suitable rank conditions, switching control strategies exist and can be built in a systematic way. The proof is based on building a new functional by the adjoint system whose minimizers are the switching controls.

2019  Impact Factor: 0.857