Optimal control of the one-dimensional periodic wave equation (original) (raw)
Second order conditions for periodic optimal control problems
Control and Cybernetics, 2005
Abstract: This paper concerns second order sufficient condi-tions of optimality, involving the Riccati equation, for optimal con-trol problems with periodic boundary conditions. The problems con-sidered involve no pathwise constraints and are 'regular', in the sense that the ...
Optimal control for a vibrating string with variable axial load and damping gain
We consider an optimal control problem for a vibrating string with multiplicative distributive control. Multiplicative control is the coefficient u(x, t) of velocity term y t . Using multiplicative control, we bring the state solution to the desired profile under a quadratic cost of control. We prove the existence of optimal control. Criterion for characterization of the unique optimal control is provided in the form of optimality system.
Mathematics and Computers in Simulation, 2008
In many real-life applications of optimal control problems with constraints in form of partial differential equations (PDEs), hyperbolic equations are involved which typically describe transport processes. Since hyperbolic equations usually propagate discontinuities of initial or boundary conditions into the domain on which the solution lives or can develop discontinuities even in the presence of smooth data, problems of this type constitute a severe challenge for both theory and numerics of PDE constrained optimization.
On solution of an optimal control problem governed by a linear wave equation
New Trends in Mathematical Science
This paper studies the minimization problem governed by a wave equation with homogeneous Neumann boundary condition and where the control function is a initial velocity of the system. We give necessary conditions for the existence and uniqueness of the optimal solution. We get the Frechet derivation of the cost functional via the solution of the corresponding adjoint problem. We construct a minimizing sequence and show that the limit of the minimizing sequence is the solution of the optimal control problem.
Lp-Optimal Boundary Control for the Wave Equation
Siam Journal on Control and Optimization, 2005
We study problems of boundary controllability with minimal L p -norm (p ∈ [2, ∞]) for the one-dimensional wave equation, where the state is controlled at both boundaries through Dirichlet or Neumann conditions. The problem is to reach a given terminal state and velocity in a given finite time, while minimizing the L p -norm of the controls. We give necessary and sufficient conditions for the solvability of this problem. We show as follows how this infinite-dimensional optimization problem can be transformed into a problem which is much simpler: The feasible set of the transformed problem is described by a finite number of simple pointwise equality constraints for the control function in the Dirichlet case while, in the Neumann case, an additional integral equality constraint appears. We provide explicit complete solutions of the problems for all p ∈ [2, ∞] in the Dirichlet case and solutions for some typical examples in the Neumann case.
Lp-Optimal Boundary Control for the Wave Equation
Siam Journal on Control and Optimization, 2005
We study problems of boundary controllability with minimal L p -norm (p ∈ [2, ∞]) for the one-dimensional wave equation, where the state is controlled at both boundaries through Dirichlet or Neumann conditions. The problem is to reach a given terminal state and velocity in a given finite time, while minimizing the L p -norm of the controls. We give necessary and sufficient conditions for the solvability of this problem. We show as follows how this infinite-dimensional optimization problem can be transformed into a problem which is much simpler: The feasible set of the transformed problem is described by a finite number of simple pointwise equality constraints for the control function in the Dirichlet case while, in the Neumann case, an additional integral equality constraint appears. We provide explicit complete solutions of the problems for all p ∈ [2, ∞] in the Dirichlet case and solutions for some typical examples in the Neumann case.
On optimal periodic solution of differential equations
this paper dedicated to the construction of solution of a three time scale periodic singular perturbed non-linear quadratic optimal control problem by using the direct method. The algorithm of the method is the direct substitution of the postulated asymptotic expansion of the solution of the problem and then by the conditions of the problem we constructed of a series of problem and find terms of the asymptotic. We find the solution by using the Hamilton's function and maximum principle.