We consider the optimal control of general heat governed by an operator depend on an unknown parameter and with missing boundary condition. Using the notion of no-regret and low-regret control we prove that we can bring the average of the state of our model to a desired state. Then by means of Euler-Lagrange first order optimality condition, we expressed the optimal control in term of average of an appropriate adjoint state that we characterize by an optimality system. The main tools are the Lebesgue dominated convergence theorem and an appropriate Hilbert space endowed with a norm containing the average of the state.

The paper is devoted to the Stackelberg control of a linear parabolic equation with missing initial condition. The strategy involves two controls called follower and leader. The objective of the follower is to bring the state to a desired state while the leader has to bring the system to rest at the final time. The results are obtained by means of Fenchel–Legendre transform and appropriate Carleman inequalities.

No abstract

In this article, we establish some new properties of the two-parameter Mittag-Leffler function and use them to prove that, mild solutions of the evolution equation) -asymptotically periodic, where A is the generator of a strongly continuous semigroup {T ( )}  0 (which is exponentially stable) on a Banach space X and C D  t denotes the Caputo fractional derivative of order 0 <  1 . We further establish an existence and uniqueness result for optimal (,c) -asymptotically periodic mild solution if X is a uniformly convex Banach space.