ISSN:

2156-8472

eISSN:

2156-8499

## Mathematical Control & Related Fields

March 2015 , Volume 5 , Issue 1

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2015, 5(1): 1-30
doi: 10.3934/mcrf.2015.5.1

*+*[Abstract](2188)*+*[PDF](488.2KB)**Abstract:**

We consider single-observed cascade systems of hyperbolic equations. We first consider the class of bounded operators that satisfy a non negativity property $(NNP)$. Within this class, we give a necessary and sufficient condition for observability of the cascade system by a single observation. We further show that if the coupling operator does not satisfy $(NNP)$ (contrarily to [5], or also e.g.[3,4] for symmetrically coupled systems), the usual observability inequality through a single component may still occur in a general framework, under some smallness conditions, but it may also be violated. When the coupling operator is a multiplication operator, $(NNP)$ is violated whenever the coupling coefficient changes sign in the spatial domain. We give explicit constructive examples of such coupling operators for which unique continuation may fail for an infinite dimensional set of initial data, that we characterize explicitly. We also exhibit examples of couplings and initial data for which the observability inequality holds but in weaker norms. These examples extend to parabolic systems. Finally, we show that the two-level energy method [1,2] which involves different levels of energies for the observed and unobserved component, may involve the

*same*levels of energies of these respective components, if the differential order of the coupling is higher (operating here through velocities instead of displacements). We further give an application to controlled systems coupled in velocities. This shows that the answer to observability and unique continuation questions for single-observed cascade systems is much more involved in the case of coupling operators that violate $(NNP)$ or of higher order coupling operators, and that the mathematical properties of the coupling operator greatly influence the dynamics of the observed system even though it operates through lower order differential terms. We indicate several extensions and future directions of research.

2015, 5(1): 31-53
doi: 10.3934/mcrf.2015.5.31

*+*[Abstract](2433)*+*[PDF](571.1KB)**Abstract:**

In this paper necessary and sufficient conditions of $L^\infty$-controllability and approximate $L^\infty$-controllability are obtained for the control system $ w_{tt}=\frac{1}{\rho} (k w_x)_x+\gamma w$, $w(0,t)=u(t)$, $x>0$, $t\in(0,T)$. Here $\rho$, $k$, and $\gamma$ are given functions on $[0,+\infty)$; $u\in L^\infty(0,\infty)$ is a control; $T>0$ is a constant. These problems are considered in special modified spaces of the Sobolev type introduced and studied in the paper. The growth of distributions from these spaces is associated with the equation data $\rho$ and $k$. Using some transformation operator introduced and studied in the paper, we see that this control system replicates the controllability properties of the auxiliary system $ z_{tt}=z_{xx}-q^2z$, $z(0,t)=v(t)$, $x>0$, $t\in(0,T)$, and vise versa. Here $q\ge0$ is a constant and $v\in L^\infty(0,\infty)$ is a control. Necessary and sufficient conditions of controllability for the main system are obtained from the ones for the auxiliary system.

2015, 5(1): 55-71
doi: 10.3934/mcrf.2015.5.55

*+*[Abstract](2377)*+*[PDF](1825.6KB)**Abstract:**

The paper gives a characterization of the uniform robust domain of attraction for a finite non-linear controlled system subject to perturbations and state constraints. We extend the Zubov approach to characterize this domain by means of the value function of a suitable infinite horizon state-constrained control problem which at the same time is a Lyapunov function for the system. We provide associated Hamilton-Jacobi-Bellman equations and prove existence and uniqueness of the solutions of these generalized Zubov equations.

2015, 5(1): 73-96
doi: 10.3934/mcrf.2015.5.73

*+*[Abstract](2810)*+*[PDF](530.6KB)**Abstract:**

In this paper we study an optimal control problem (OCP) associated to a linear elliptic equation on a bounded domain $\Omega$. The matrix-valued coefficients $A$ of such systems is our control in $\Omega$ and will be taken in $L^2(\Omega;\mathbb{R}^{N\times N})$ which in particular may comprises the case of unboundedness. Concerning the boundary value problems associated to the equations of this type, one may exhibit non-uniqueness of weak solutions--- namely, approximable solutions as well as another type of weak solutions that can not be obtained through the $L^\infty$-approximation of matrix $A$. Following the direct method in the calculus of variations, we show that the given OCP is well-possed and admits at least one solution. At the same time, optimal solutions to such problem may have a singular character in the above sense. In view of this we indicate two types of optimal solutions to the above problem: the so-called variational and non-variational solutions, and show that some of that optimal solutions can not be attainable through the $L^\infty$-approximation of the original problem.

2015, 5(1): 97-139
doi: 10.3934/mcrf.2015.5.97

*+*[Abstract](4661)*+*[PDF](614.2KB)**Abstract:**

A linear-quadratic (LQ, for short) optimal control problem is considered for mean-field stochastic differential equations with constant coefficients in an infinite horizon. The stabilizability of the control system is studied followed by the discussion of the well-posedness of the LQ problem. The optimal control can be expressed as a linear state feedback involving the state and its mean, through the solutions of two algebraic Riccati equations. The solvability of such kind of Riccati equations is investigated by means of semi-definite programming method.

2015, 5(1): 141-163
doi: 10.3934/mcrf.2015.5.141

*+*[Abstract](2264)*+*[PDF](600.4KB)**Abstract:**

We construct a patchy feedback for a general control system on $\mathbb{R}^d$ which realizes practical stabilization to a target set $\Sigma$, when the dynamics is constrained to a given set of states $S$. The main result is that $S$--constrained asymptotically controllability to $\Sigma$ implies the existence of a discontinuous practically stabilizing feedback. Such a feedback can be constructed in ``patchy'' form, a particular class of piecewise constant controls which ensure the existence of local Carathéodory solutions to any Cauchy problem of the control system and which enjoy good robustness properties with respect to both measurement errors and external disturbances.

2015, 5(1): 165-176
doi: 10.3934/mcrf.2015.5.165

*+*[Abstract](2117)*+*[PDF](338.8KB)**Abstract:**

This paper is addressed to a quantitative internal unique continuation property for stochastic parabolic equations, i.e., we show that each of their solutions can be determined by the observation on any nonempty open subset of the whole region in which the equations evolve. The proof is based on a global Carleman estimate.

2015, 5(1): 177-189
doi: 10.3934/mcrf.2015.5.177

*+*[Abstract](2379)*+*[PDF](352.8KB)**Abstract:**

In this paper we establish a global Carleman estimate for the fourth order Schrödinger equation with potential posed on a $1-d$ finite domain. The Carleman estimate is used to prove the Lipschitz stability for an inverse problem consisting in recovering a stationary potential in the Schrödinger equation from boundary measurements.

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