Thank you, Toby, for the invitation. And it's my honor that I can be here to give a talk

about first-order numerical algorithms for some optimal control problems with the contents.

And actually, this talk is a brief summary of my work during the PT study.

First, I will present some preliminaries on optimal control problems, and then to expose

the main ideas on the algorithmic design, we will focus on the following two kinds of

optimal control problems.

Okay, the PDE constraint, optimal control problems with the constraints, and embedding

your optimal control over an advection reaction diffusion equation.

And finally, there are some conclusions.

Optimal control aims at controlling a given system over a period of time, such as that

a certain goal is achieved.

The system can be mathematically described by ODEs, PDEs, integral equations, difference

equations, etc.

And here we focus on some PDE-reduced optimal control problems.

Tactically, an optimal control problem with PDE constraints can be represented as this

model.

And here, u and y are binary spaces, and uad and yad are closed convex sites.

D is the object function to be minimized.

And E, like zero, represents a PDE or a system of coupled PDEs.

The state variable y describes the state of the considered system, and the control variable

u is a parameter that shall be adapted in an optimal way.

And here the control constraint u belongs to uad, and the state constraint y belongs

to yad, that describes some physical restrictions and realistic requirements.

And there are many works for theoretical analysis of the existence and regularity of optimal

solution, numerical discretization schemes, and some applications perceptive.

And here we invite algorithmic design for this problem.

Generally, it is not easy to find an efficient solver for solving optimal control problems

with PDE constraints.

And in particular, from PDE perspectives, the models are generally complex, and solving

the involved PDEs is already not easy.

And the results for PDEs, including the error estimation or error solver, cannot be extended

directly to the optimal control settings.

On the other hand, from optimization perspectives, there are the high dimensionality after the

discretization of PDEs, especially for the time dependent cases, and extremely ill-conditioned

linear systems, and some theoretical obstacles.

So simply combining of the PDE solvers and optimization algorithms does not work.

And specific structures and properties of the model should be considered deliberately.

Okay, next we consider some PDE constraints and optimal control problems with control

constraints.

Okay.

First, we focus on some PDE-contrained optimal control problems with control constraints

that can be unified as this model.

And here the constant alpha, rather than zero, is a regularization parameter.

And the function yd is a given target.

And here s is a solution operator associated with some linear PDEs.

And the next most complex function, theta, represents some additional box of sparsity

to contain the control variable.

For example, theta can be the indicator function of the over an automatable site of u, or the

one norm of u to promote the sparsity of u.

And to solve this model, there are various simultaneous Newton methods and interior point