Optimal time delays in a class of reaction-diffusion equations

ABSTRACT

A class of semilinear parabolic reaction diffusion equations with multiple time delays is considered. These time delays and corresponding weights are to be optimized such that the associated solution of the delay equation is the best approximation of a desired state function. The differentiability of the mapping is proved that associates the solution of the delay equation to the vector of weights and delays. Based on an adjoint calculus, first-order necessary optimality conditions are derived. Numerical test examples show the applicability of the concept of optimizing time delays.

KEYWORDS:

• Semilinear parabolic equation
• multiple time delays
• Pyragas type feedback
• optimization
• learning controller

AMS CLASSIFICATIONS:

• 49K20
• 49M05
• 35K58

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

M. Mateos  http://orcid.org/0000-0003-3100-412X

Additional information

Funding

The first two authors were partially supported by Spanish Ministerio de Economía y Competitividad under research projects MTM2014-57531-P and MTM2017-83185-P. The third author was supported by Deutsche Forschungsgemeinschaft, Collaborative research center SFB 910, TU Berlin, project B6.