35 research outputs found
Optimal control of fractional semilinear PDEs
In this paper we consider the optimal control of semilinear fractional PDEs
with both spectral and integral fractional diffusion operators of order
with . We first prove the boundedness of solutions to both
semilinear fractional PDEs under minimal regularity assumptions on domain and
data. We next introduce an optimal growth condition on the nonlinearity to show
the Lipschitz continuity of the solution map for the semilinear elliptic
equations with respect to the data. We further apply our ideas to show
existence of solutions to optimal control problems with semilinear fractional
equations as constraints. Under the standard assumptions on the nonlinearity
(twice continuously differentiable) we derive the first and second order
optimality conditions
A note on semilinear fractional elliptic equation: analysis and discretization
In this paper we study existence, regularity, and approximation of solution
to a fractional semilinear elliptic equation of order . We
identify minimal conditions on the nonlinear term and the source which leads to
existence of weak solutions and uniform -bound on the solutions. Next
we realize the fractional Laplacian as a Dirichlet-to-Neumann map via the
Caffarelli-Silvestre extension. We introduce a first-degree tensor product
finite elements space to approximate the truncated problem. We derive a priori
error estimates and conclude with an illustrative numerical example
Exponential Turnpike property for fractional parabolic equations with non-zero exterior data
We consider averages convergence as the time-horizon goes to infinity of
optimal solutions of time-dependent optimal control problems to optimal
solutions of the corresponding stationary optimal control problems. Control
problems play a key role in engineering, economics and sciences. To be more
precise, in climate sciences, often times, relevant problems are formulated in
long time scales, so that, the problem of possible asymptotic behaviors when
the time-horizon goes to infinity becomes natural. Assuming that the controlled
dynamics under consideration are stabilizable towards a stationary solution,
the following natural question arises: Do time averages of optimal controls and
trajectories converge to the stationary optimal controls and states as the
time-horizon goes to infinity? This question is very closely related to the
so-called turnpike property that shows that, often times, the optimal
trajectory joining two points that are far apart, consists in, departing from
the point of origin, rapidly getting close to the steady-state (the turnpike)
to stay there most of the time, to quit it only very close to the final
destination and time. In the present paper we deal with heat equations with
non-zero exterior conditions (Dirichlet and nonlocal Robin) associated with the
fractional Laplace operator (). We prove the turnpike
property for the nonlocal Robin optimal control problem and the exponential
turnpike property for both Dirichlet and nonlocal Robin optimal control
problems