Turnpike property for functionals involving L1−norm

Introduction

We introduce the following notation: , ,

and the correspondent norms and . Moreover, we define the norms

We want to study the following optimal control problem:

where is defined by

and is the solution of the PDE given by

Notice that, by integration by parts, , where is solution of the adjoint equation:

Sparse control: ()

The stationary problem

Numerical algorithm

In order to compute a numerical solution of problem , after a discretization by finite differences, we use a prox-prox splitting: first write the state as , then

• Proximal-point step:
•

• Proximal-point step:
•

Remark: Notice that, when , the solution of is simply given by

Evolutionary problem

Optimality conditions

Define the classical Lagrangian

By integration by parts, we have

Deriving with respect to the three variables , we obtain the optimality system:

where the relation between the optimal control and the dual state is given by

The latter is equivalent to

where the operator of is defined by

Finally,

Numerical algorithm

In order to compute a numerical solution of problem , after a discretization by finite differences, we use a grad-prox splitting:

•

where

and

• Proximal-point step:
•

Remarks: Another possibility is to include the term in the proximal step.
Notice that, for

then is Lipschitz continuous. Indeed, for (), then

where

and

By linearity and solve the same equations with right-hand-sides and , respectively. Then

where we defined

In order the prox-grad method to converge, the restriction on the step size is given by

Sparse state: ()

The stationary problem

Optimality conditions

Finally, we obtain a single equation in the dual variable :

Numerical algorithm

In order to compute a numerical solution of problem , after a discretization by finite differences, we use a prox-prox splitting on the Augmented Energy: first write the state as , then

• Proximal-point step:
•

• Proximal-point step:
•

where we defined

Remark: Notice that again, when , the solution of is simply given by

Evolutionary problem
Optimality conditions

Define the classical Lagrangian

By integration by parts, we have

Deriving with respect to the three variables , we obtain the optimality system:

where the relation between the optimal control and the dual state is given by

The adjoint equation is equivalent to

Finally,

Numerical algorithm

In order to compute a numerical solution of problem , after a discretization by finite differences, we use a grad-prox splitting on the following Augmented Energy:

Then,

•

where

and

• Proximal-point step:
•

where we defined

Remark: Another possibility is to consider

Computational experiments

In the following, we present the setting for the numerical experiments.

• Spacial domain: ;
• Time interval: , with ;
• Weight-parameters: , , and ;
• Trajectory target:

where , ;

• Control operator: for and ,

• is the finite difference discretization of ;
• Numerical grid: in space, in time.