This paper is devoted to the problem of partial asymptotic null-controllability of control sys-tems governed by ordinary differential equations,subjected to possibly mixed state-input con-straints.Using Lyapunov funct...This paper is devoted to the problem of partial asymptotic null-controllability of control sys-tems governed by ordinary differential equations,subjected to possibly mixed state-input con-straints.Using Lyapunov functions within the framework of viability theory,feedback controls are designed in such a way a part of system’s state can be driven to the origin asymptotically,taking into account the mixed constraints.By using Michael selection theorem,the existence of such controls is proved,in the case of convex constraints,and their expressions are given as continu-ous selections of an appropriate constructed multifunction.Finally,two examples are processed numerically in order to illustrate the theoretical results.展开更多
The article considers the controllability of a diffusion equation with fractional integro-differential expressions.We prove that the resulting equation is nullcontrollable in arbitrary small time.Our method reduces es...The article considers the controllability of a diffusion equation with fractional integro-differential expressions.We prove that the resulting equation is nullcontrollable in arbitrary small time.Our method reduces essentially to the study of classical moment problems.展开更多
The advection-diffusion equation y_t~ε-εy_(xx)~ε+ M y_x~ε= 0,(x, t) ∈(0, 1) ×(0, T) is null controllable for any strictly positive values of the diffusion coefficient ε and of the controllability time T. We...The advection-diffusion equation y_t~ε-εy_(xx)~ε+ M y_x~ε= 0,(x, t) ∈(0, 1) ×(0, T) is null controllable for any strictly positive values of the diffusion coefficient ε and of the controllability time T. We discuss here the behavior of the cost of control when the coefficient ε goes to zero, according to the values of T. It is actually known that this cost is uniformly bounded with respect to ε if T is greater than a minimal time T_M, with T_M in the interval [1, 2×3^(1/2)]/M for M > 0 and in the interval [2×2^(1/2), 2(1 +3^(1/2))]/|M | for M < 0. The exact value of TM is however unknown.We investigate in this work the determination of the minimal time T_M employing two distincts but complementary approaches. In a first one, we numerically estimate the cost of controllability, reformulated as the solution of a generalized eigenvalue problem for the underlying control operator, with respect to the parameter T and ε. This allows notably to exhibit the structure of initial data leading to large costs of control. At the practical level, this evaluation requires the non trivial and challenging approximation of null controls for the advection-diffusion equation. In the second approach, we perform an asymptotic analysis, with respect to the parameter ε, of the optimality system associated to the control of minimal L^2-norm. The matched asymptotic expansion method is used to describe the multiple boundary layers.展开更多
This paper is concerned with the bound of the cost of approximate controllability and null controllability of heat equations, i.e., the minimal Lp norm and L∞ norm of a control needed to control the system approximat...This paper is concerned with the bound of the cost of approximate controllability and null controllability of heat equations, i.e., the minimal Lp norm and L∞ norm of a control needed to control the system approximately or a control needed to steer the state of the system to zero. The methods we use combine observability inequalities, energy estimates for heat equations and the dual theory.展开更多
In this paper, we establish a bang-bang principle of time optimal controls for a controlled parabolic equation of fractional order evolved in a bounded domain Ω of R^n, with a controller w to be any given nonempty op...In this paper, we establish a bang-bang principle of time optimal controls for a controlled parabolic equation of fractional order evolved in a bounded domain Ω of R^n, with a controller w to be any given nonempty open subset of Ω. The problem is reduced to a new controllability property for this equation, i.e. the null controllability of the system at any given time T 〉 0 when the control is restricted to be active in ω× E, where E is any given subset of [0, T] with positive （Legesgue） measure. The desired controllability result is established by means of a sharp observability estimate on the eigenfunctions of the Dirichlet Laplacian due to Lebeau and Robbiano, and a delicate result in the measure theory due to Lions.展开更多
In the paper, the null interior controllability for a fourth order parabolic equation is obtained. The method is based on Lebeau-Rabbiano inequality which is a quantitative unique continuation property for the sum of ...In the paper, the null interior controllability for a fourth order parabolic equation is obtained. The method is based on Lebeau-Rabbiano inequality which is a quantitative unique continuation property for the sum of eigenfunctions of the Laplacian.展开更多
The aim of this work is to improve the minimum time of null controllability of the 1D heat equation by using the notion of strategic zone actuators. In fact, motivated by the work of Khodja on the null controllability...The aim of this work is to improve the minimum time of null controllability of the 1D heat equation by using the notion of strategic zone actuators. In fact, motivated by the work of Khodja on the null controllability of the heat equation and of El Jai on the controllability by the use of strategic zone actuators, we managed, in this work, to improve the minimal time of null controllability to the 1D heat equation. However, the restrictions and difficulties to establish the inequality of coercivity of the parabolic operator, require to seek other methods of internal control. Thus in this paper, a mixed method combining the method of moments and the notion of strategic profile was used to find a better minimal time of null controllability of the 1D heat equation.展开更多
文摘This paper is devoted to the problem of partial asymptotic null-controllability of control sys-tems governed by ordinary differential equations,subjected to possibly mixed state-input con-straints.Using Lyapunov functions within the framework of viability theory,feedback controls are designed in such a way a part of system’s state can be driven to the origin asymptotically,taking into account the mixed constraints.By using Michael selection theorem,the existence of such controls is proved,in the case of convex constraints,and their expressions are given as continu-ous selections of an appropriate constructed multifunction.Finally,two examples are processed numerically in order to illustrate the theoretical results.
基金Supported by National Natural Science Foundation of China(No.11261024).
文摘The article considers the controllability of a diffusion equation with fractional integro-differential expressions.We prove that the resulting equation is nullcontrollable in arbitrary small time.Our method reduces essentially to the study of classical moment problems.
文摘The advection-diffusion equation y_t~ε-εy_(xx)~ε+ M y_x~ε= 0,(x, t) ∈(0, 1) ×(0, T) is null controllable for any strictly positive values of the diffusion coefficient ε and of the controllability time T. We discuss here the behavior of the cost of control when the coefficient ε goes to zero, according to the values of T. It is actually known that this cost is uniformly bounded with respect to ε if T is greater than a minimal time T_M, with T_M in the interval [1, 2×3^(1/2)]/M for M > 0 and in the interval [2×2^(1/2), 2(1 +3^(1/2))]/|M | for M < 0. The exact value of TM is however unknown.We investigate in this work the determination of the minimal time T_M employing two distincts but complementary approaches. In a first one, we numerically estimate the cost of controllability, reformulated as the solution of a generalized eigenvalue problem for the underlying control operator, with respect to the parameter T and ε. This allows notably to exhibit the structure of initial data leading to large costs of control. At the practical level, this evaluation requires the non trivial and challenging approximation of null controls for the advection-diffusion equation. In the second approach, we perform an asymptotic analysis, with respect to the parameter ε, of the optimality system associated to the control of minimal L^2-norm. The matched asymptotic expansion method is used to describe the multiple boundary layers.
基金Supported by Key Project of Chinese Ministry of Education (No. i08046)National Natural Science Foundation of China (Nos. 10601010,10471021)Grant NENU-STC07007
文摘This paper is concerned with the bound of the cost of approximate controllability and null controllability of heat equations, i.e., the minimal Lp norm and L∞ norm of a control needed to control the system approximately or a control needed to steer the state of the system to zero. The methods we use combine observability inequalities, energy estimates for heat equations and the dual theory.
基金Partially supported by National Natural Science Foundation of China (Grant No. 10525105)the NCET of China (Grant No. 04-0882)
文摘In this paper, we establish a bang-bang principle of time optimal controls for a controlled parabolic equation of fractional order evolved in a bounded domain Ω of R^n, with a controller w to be any given nonempty open subset of Ω. The problem is reduced to a new controllability property for this equation, i.e. the null controllability of the system at any given time T 〉 0 when the control is restricted to be active in ω× E, where E is any given subset of [0, T] with positive （Legesgue） measure. The desired controllability result is established by means of a sharp observability estimate on the eigenfunctions of the Dirichlet Laplacian due to Lebeau and Robbiano, and a delicate result in the measure theory due to Lions.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 10671040, 10831007, 10801041)the National Excellent Doctoral Dissertation of China (Grant No. 200522)the New Century Excellent Talents in University (Grant No. 06-0359)
文摘In the paper, the null interior controllability for a fourth order parabolic equation is obtained. The method is based on Lebeau-Rabbiano inequality which is a quantitative unique continuation property for the sum of eigenfunctions of the Laplacian.
文摘The aim of this work is to improve the minimum time of null controllability of the 1D heat equation by using the notion of strategic zone actuators. In fact, motivated by the work of Khodja on the null controllability of the heat equation and of El Jai on the controllability by the use of strategic zone actuators, we managed, in this work, to improve the minimal time of null controllability to the 1D heat equation. However, the restrictions and difficulties to establish the inequality of coercivity of the parabolic operator, require to seek other methods of internal control. Thus in this paper, a mixed method combining the method of moments and the notion of strategic profile was used to find a better minimal time of null controllability of the 1D heat equation.
基金supported by the National Nature Science Foundation of China(Nos.11371084,11231007 and 11322110)the National Basic Research Program of China("973"Program)(No.2011CB808002)+3 种基金the Program for New Century Excellent Talents in Chinese University(NCET–12–0812)the Foundation for the Author of National Excellent Doctoral Dissertation of China(No.201213)the project MTM2011–29306 of the Spanish Science and Innovation Ministrythe Chang Jiang Scholars Program(from the Chinese Education Ministry)
