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Mathematical Control & Related Fields

2015 , Volume 5 , Issue 2

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Feedback optimal control for stochastic Volterra equations with completely monotone kernels
Fulvia Confortola and Elisa Mastrogiacomo
2015, 5(2): 191-235 doi: 10.3934/mcrf.2015.5.191 +[Abstract](66) +[PDF](668.5KB)
In this paper we are concerned with a class of stochastic Volterra integro-differential problems with completely monotone kernels, where we assume that the noise enters the system when we introduce a control. We start by reformulating the state equation into a semilinear evolution equation which can be treated by semigroup methods. The application to optimal control provides other interesting results and requires a precise description of the properties of the generated semigroup.
    The first main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The main technical point consists in the differentiability of the BSDE associated with the reformulated equation with respect to its initial datum $x$.
Lookback option pricing for regime-switching jump diffusion models
Zhuo Jin and Linyi Qian
2015, 5(2): 237-258 doi: 10.3934/mcrf.2015.5.237 +[Abstract](91) +[PDF](511.8KB)
In this paper, we will introduce a numerical method to price the European lookback floating strike put options where the underlying asset price is modeled by a generalized regime-switching jump diffusion process. In the Markov regime-switching model, the option value is a solution of a coupled system of nonlinear integro-differential partial differential equations. Due to the complexity of regime-switching model, the jump process involved, and the nonlinearity, closed-form solutions are virtually impossible to obtain. We use Markov chain approximation techniques to construct a discrete-time Markov chain to approximate the option value. Convergence of the approximation algorithms is proved. Examples are presented to demonstrate the applicability of the numerical methods.
Largest space for the stabilizability of the linearized compressible Navier-Stokes system in one dimension
Debanjana Mitra, Mythily Ramaswamy and Jean-Pierre Raymond
2015, 5(2): 259-290 doi: 10.3934/mcrf.2015.5.259 +[Abstract](65) +[PDF](532.0KB)
In this paper we determine the largest space in which the linearized compressible Navier-Stokes system in one dimension, with periodic boundary conditions, is stabilizable with any prescribed exponential decay rate, by an interior control acting only in the velocity equation. As a consequence, it also follows that this largest space for the stabilizability with any prescribed exponential decay rate is also the largest one for the null controllability of the same system.
A note on optimality conditions for optimal exit time problems
Luong V. Nguyen
2015, 5(2): 291-303 doi: 10.3934/mcrf.2015.5.291 +[Abstract](49) +[PDF](330.8KB)
In this note, we obtain some optimality conditions for optimal control problems with exit time similar to those obtained in [Cannarsa, Pignotti and Sinestrari, Discrete Contin. Dynam. Systems 6 (2000), 975 - 997] without requiring an assumption on the Hamiltonian.
Stability and controllability of a wave equation with dynamical boundary control
Bopeng Rao, Laila Toufayli and Ali Wehbe
2015, 5(2): 305-320 doi: 10.3934/mcrf.2015.5.305 +[Abstract](105) +[PDF](389.5KB)
In this work, we consider the stabilization and the exact controllability of a wave equation with dynamical boundary control. We first prove the strong stability of the system and establish a polynomial decay rate for smooth solutions. We next show the exact controllability by means of a singular dynamical boundary control.
Exponential stability of a joint-leg-beam system with memory damping
Qiong Zhang
2015, 5(2): 321-333 doi: 10.3934/mcrf.2015.5.321 +[Abstract](94) +[PDF](413.0KB)
In this paper, we consider a system for combined axial and transverse motions of two viscoelastic Euler-Bernoulli beams connected through two legs to a joint. This model comes from rigidizable and inflatable space structures. First, the exponential stability of the joint-leg-beam system is obtained when both beams are subject to viscoelastic damping and memory kernels satisfy reasonable assumptions. Then, we show the lack of uniform decay of the coupled system when only one beam is assumed to have a memory damping and the second beam has no damping.
Global controllability and stabilizability of Kawahara equation on a periodic domain
Xiangqing Zhao and Bing-Yu Zhang
2015, 5(2): 335-358 doi: 10.3934/mcrf.2015.5.335 +[Abstract](71) +[PDF](494.4KB)
In this paper we study controllability and stabilizability of a class of distributed parameter control system described by the Kawahara equation posed on a periodic domain $\mathbb{T}$ with internal control acting on a sub-domain $\omega $ of $\mathbb{T}$. Earlier in [42], aided by Bourgain smoothing property of the system, we showed that the system is locally exactly controllable and exponentially stabilizable. In this paper, helped further by certain properties of propagation of compactness and regularity in Bourgain spaces for the solutions of the associated linear system, we show that the system is globally exactly controllable and globally exponentially stabilizable.
Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems
Guangliang Zhao, Fuke Wu and George Yin
2015, 5(2): 359-376 doi: 10.3934/mcrf.2015.5.359 +[Abstract](99) +[PDF](489.7KB)
To treat networked systems involving uncertainty due to randomness with both continuous dynamics and discrete events, this paper focuses on diffusions modulated by a continuous-time Markov chain. In our paper [19], we considered ordinary differential equations with Markovian switching. This paper further treats more complex cases, namely, stochastic differential equations with Markovian switching. Our goal is to stabilize the systems under consideration. One of the difficulties is that the systems grow much faster than the allowable rates in the literature of stochastic differential equations. As a result, the underlying systems have finite explosion time. To overcome the difficulties, we develop feedback controls to extend the local solutions to global solutions and to stabilize the resulting systems. The feedback controls are Brownian type of perturbations. We establish the existence of global solution, prove the stability of the resulting systems, obtain boundedness in probability as $t\to\infty$, and provide sufficient conditions for almost sure stability. Then we present numerical examples to illustrate the main results.

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