Optimal control of the one-dimensional periodic wave equation (original) (raw)

Numerical Optimal Control of The Wave Equation

2020

In this paper, we present a spectral method for approximating the boundary optimal control problems of a well-known wave equation by the linear optimal control problems. The method is based upon constructing the Mth degree interpolation polynomials, using Chebyshevs nodes, to approximate the wave equation. Necessary conditions for optimal control functions are obtained by using the Pontryagin's maximum principle. Moreover, the control parameterization enhancing technique (CPET) is used to obtain the piecewise constant sub-optimal control functions. Finally, the efficiency of the proposed method is confirmed by a numerical example.

Optimal Control of Linear Lower Order Non-Dispersive Waves

2012

We can majorly distinguish two main classes of waves, namely, dispersive and nondispersive waves. The latter class, also termed the hyperbolic waves, are so-called since they can be formulated in terms of hyperbolic partial differential equations. Generally, there are waves that exhibit both types of behaviours but dispersive waves are not classified as easily as non-dispersive. This paper considers the optimal control of lower order wave which has its essential role in applications. For example, the higher order waves, whose optimal control was studied by Reju [1, 2], often carry the "first signal" when combined with lower order waves, but the main disturbance or propagation travels with the lower order waves as confirmed by some of our simulated results in this work when compared with the higher order control problem of Reju [1, 2 ]. A Hamiltonian approach is employed to solve the model optimization problem with the resulting semi-analytical solutions computationally simulated for analysis.

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.

Identification Problem for a Wave Equation via Optimal Control

IFIP Advances in Information and Communication Technology, 1999

We approximate an identification problem by applying optimal control techniques to a Tikhonov's regularization. We seek to identify the dispersive coefficient in a wave equation and allow for the case of error or uncertainty in the observations used for the identification.

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.

Numerical optimal control of the wave equation: optimal boundary control of a string to rest in finite time

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.

OPTIMAL CONTROL FOR A CONTROLLED ILLPOSED WAVE EQUATION WITHOUT REQUIRING THE SLATER HYPOTHESIS

Ural mathematical journal , 2020

In this paper, we investigate the problem of optimal control for an ill-posed wave equation without using the extra hypothesis of Slater i.e. the set of admissible controls has a non-empty interior. Firstly, by a controllability approach, we make the ill-posed wave equation a well-posed equation with some incomplete data initial condition. The missing data requires us to use the no-regret control notion introduced by Lions to control distributed systems with incomplete data. After approximating the no-regret control by a low-regret control sequence, we characterize the optimal control by a singular optimality system.