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Volume 1, 2011

Mathematical Control & Related Fields

2011 , Volume 1 , Issue 4

Dedicated to Professor Yutaka Takahashi on the occasion of his 60th birthday

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Indirect stabilization of weakly coupled systems with hybrid boundary conditions
Fatiha Alabau-Boussouira, Piermarco Cannarsa and Roberto Guglielmi
2011, 1(4): 413-436 doi: 10.3934/mcrf.2011.1.413 +[Abstract](96) +[PDF](496.4KB)
Abstract:
We investigate stability properties of indirectly damped systems of evolution equations in Hilbert spaces, under new compatibility assumptions. We prove polynomial decay for the energy of solutions and optimize our results by interpolation techniques, obtaining a full range of power-like decay rates. In particular, we give explicit estimates with respect to the initial data. We discuss several applications to hyperbolic systems with hybrid boundary conditions, including the coupling of two wave equations subject to Dirichlet and Robin type boundary conditions, respectively.
Pathwise Taylor expansions for Itô random fields
Rainer Buckdahn, Ingo Bulla and Jin Ma
2011, 1(4): 437-468 doi: 10.3934/mcrf.2011.1.437 +[Abstract](138) +[PDF](702.2KB)
Abstract:
In this paper we study the pathwise stochastic Taylor expansion, in the sense of our previous work [3], for a class of Itô-type random fields in which the diffusion part is allowed to contain both the random field itself and its spatial derivatives. Random fields of such an "self-exciting" type particularly contains the fully nonlinear stochastic PDEs of curvature driven diffusion, as well as certain stochastic Hamilton-Jacobi-Bellman equations. We introduce the new notion of "$n$-fold" derivatives of a random field, as a fundamental device to cope with the special self-exciting nature. Unlike our previous work [3], our new expansion can be defined around any random time-space point (τ,ξ), where the temporal component τ does not even have to be a stopping time. Moreover, the exceptional null set is independent of the choice of the random point (τ,ξ). As an application, we show how this new form of pathwise Taylor expansion could lead to a different treatment of the stochastic characteristics for a class of fully nonlinear SPDEs whose diffusion term involves both the solution and its gradient, and hence lead to a definition of the stochastic viscosity solution for such SPDEs, which is new in the literature, and potentially of essential importance in stochastic control theory.
Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction
Martin Gugat and Markus Dick
2011, 1(4): 469-491 doi: 10.3934/mcrf.2011.1.469 +[Abstract](140) +[PDF](368.5KB)
Abstract:
We consider the isothermal Euler equations with friction that model the gas flow through pipes. We present a method of time-delayed boundary feedback stabilization to stabilize the isothermal Euler equations locally around a given stationary subcritical state on a finite time interval. The considered control system is a quasilinear hyperbolic system with a source term. For this system we introduce a Lyapunov function with delay terms and develop time-delayed boundary controls for which the Lyapunov function decays exponentially with time. We present the stabilization method for a single gas pipe and for a star-shaped network of pipes.
Optimal control of a vector-host epidemics model
Qingkai Kong, Zhipeng Qiu, Zi Sang and Yun Zou
2011, 1(4): 493-508 doi: 10.3934/mcrf.2011.1.493 +[Abstract](94) +[PDF](556.4KB)
Abstract:
In this paper, a vector-host epidemic model with control measures is considered to assess the impact of control measures on the prevalence of the vector-host diseases. We incorporated mosquito-reduction strategy and host medical treatment into the model. For the basic vector-host model, we provide sufficient conditions for the local stability of the disease free equilibrium (DFE) and the sensitivity analysis for the reproduction number with respect to the model parameters. Using the optimal control theory, the optimal levels of the two controls are characterized, and then the existence and uniqueness for the optimal control pair are established. Numerical simulations are further conducted to confirm and extend the analytical results. Numerical results suggest that optimal multi-control strategy is a more beneficial choice in fighting the outbreak of vector-host diseases. For the vector-host epidemics, vector control measures should be taken prior to other measures.
Inverse source problem with a final overdetermination for a fractional diffusion equation
Kenichi Sakamoto and Masahiro Yamamoto
2011, 1(4): 509-518 doi: 10.3934/mcrf.2011.1.509 +[Abstract](165) +[PDF](316.6KB)
Abstract:
For a time fractional diffusion equation with source term, we discuss an inverse problem of determining a spatially varying function of the source by final overdetermining data. We prove that this inverse problem is well-posed in the Hadamard sense except for a discrete set of values of diffusion constants.

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