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Mathematical Control & Related Fields

June 2012 , Volume 2 , Issue 2

Special Issue dedicated to Professor Charles Pearce on the occasion of his 70th birthday

Special Issue Papers: 223-375; Regular Papers: 377-435

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Numerical approximation of an optimization problem to reduce leakage in water distribution systems
Pierre Fabrie, Elodie Jaumouillé, Iraj Mortazavi and Olivier Piller
2012, 2(2): 101-120 doi: 10.3934/mcrf.2012.2.101 +[Abstract](619) +[PDF](827.1KB)
Abstract:
Leakage represents a large part of the supplied water in Water Distribution Systems (WDS). Consequently, it is important to develop some efficient strategies to manage such a phenomenon. In this paper an improved formulation of the hydraulic network equations that incorporate pressure-dependent leakage, is presented and validated. The formulation is derived from the Navier-Stokes equations and solved using an adequate splitting method. Then, this formulation is used to study a constrained optimization problem with the objective to minimize the distributed water volume reducing the leakage. The problem is described and validated for academic case studies and real networks.
On the control of some coupled systems of the Boussinesq kind with few controls
Enrique Fernández-Cara and Diego A. Souza
2012, 2(2): 121-140 doi: 10.3934/mcrf.2012.2.121 +[Abstract](561) +[PDF](436.5KB)
Abstract:
This paper is devoted to prove the local exact controllability to the trajectories for a coupled system, of the Boussinesq kind, with a reduced number of controls. In the state system, the unknowns are the velocity field and pressure of the fluid $(\mathbf{y},p)$, the temperature $\theta$ and an additional variable $c$ that can be viewed as the concentration of a contaminant solute. We prove several results, that essentially show that it is sufficient to act locally in space on the equations satisfied by $\theta$ and $c$.
The simplest semilinear parabolic equation of normal type
Andrei Fursikov
2012, 2(2): 141-170 doi: 10.3934/mcrf.2012.2.141 +[Abstract](533) +[PDF](536.1KB)
Abstract:
The notion of semilinear parabolic equation of normal type is introduced. The structure of dynamical flow corresponding to equation of this type with periodic boundary condition is investigated. Stabilization of mentioned equation with arbitrary initial condition by start control supported in prescribed subset is constructed.
Approximate controllability of semilinear reaction diffusion equations
Hugo Leiva, Nelson Merentes and José L. Sánchez
2012, 2(2): 171-182 doi: 10.3934/mcrf.2012.2.171 +[Abstract](744) +[PDF](367.6KB)
Abstract:
In this paper we prove the approximate controllability of the a broad class of semilinear reaction diffusion equation in a Hilbert space, with application to the semilinear $n$D heat equation, the Ornstein-Uhlenbeck equation, amount others.
Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs
Ming Yan, Lili Chang and Ningning Yan
2012, 2(2): 183-194 doi: 10.3934/mcrf.2012.2.183 +[Abstract](743) +[PDF](343.0KB)
Abstract:
In this paper, we study the finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Instead of the standard error estimates under $L^2$- or $H^1$- norm, we apply the goal-oriented error estimates in order to avoid the difficulties which are generated by the nonsmoothness of the problem. We derive the a priori error estimates of the goal function, and the error bound is $O(h^2)$, which is the same as one for some well known quadratic optimal control problems governed by linear elliptic PDEs. Moreover, two kinds of practical algorithms are introduced to solve the underlying problem. Numerical experiments are provided to confirm our theoretical results.
A unified theory of maximum principle for continuous and discrete time optimal control problems
Zaidong Zhan, Shuping Chen and Wei Wei
2012, 2(2): 195-215 doi: 10.3934/mcrf.2012.2.195 +[Abstract](934) +[PDF](469.1KB)
Abstract:
Traditionally, the time domains that are widely used in mathematical descriptions are limited to real numbers for the case of continuous-time optimal control problems or to integers for the case of discrete-time optimal control problems. In this paper, based on a family of "needle variations", we derive maximum principle for optimal control problem on time scales. The results not only unify the theory of continuous and discrete optimal control problems but also conclude problems involving time domains in partly continuous and partly discrete ingredients. A simple optimal control problem on time scales is discussed in detail. Meanwhile, the results also unify the theory of some hybrid systems, for example, impulsive systems.

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