Unifying Optimal Control Problems through Constraint Optimization Problem

Question

Consider Optimization problem in the form $$\min ~ f(x) \quad \quad \text{subject to}\quad \quad F(x) \in C \subseteq Y $$

Where $F: X \to Y$ is continuously differentiable between banach spaces $X$ and $Y$ and $f$ is lipschitz function, and $C$ is closed convex set in $Y$. I'm wondering is there any class of optimal control problems fitted in above framework ?

I'm asking this, because KKT condition is available for above problem in literatures, and yet expert people in optimal control take different approach to ge necessary optimality condition! To my understanding an optimal control problem is just an Infinite dimensional optimization problem, why we can't employ one single approach to solve both.

My Guess:

I think the assumption $F$ be differentiable (in Frechet sense) is strong, and might make trouble to cover large class of optimal control problems. I'm asking this, because I found that expert people in optimal control take different approach to solve problem,

Answer 1

Sure, for example sparse control of PDEs with state constraints ($f(x) = \|x\|_{L^1}$, $F(x)$ the solution operator to your favorite PDE, and $C = \{x(t)\leq 1 \text{ for all }t\}$ -- although care needs to be taken to ensure existence of a solution).

For differentiable $f$, casting such optimal control problems for PDEs in the language of nonlinear optimization in Banach spaces is rather well-known and popular; for example, it's the approach used in the popular textbook by Fredi Tröltzsch [1]. The standard constraint qualification used is based on that by Zowe and Kurcyusz [2]. The case of non-differentiable and in particular non-convex $f$ (and $C$) in optimal control is currently a very active field, where it is a matter of preference whether people work with the (mainly equivalent) generalized derivatives of the indicator function of $C$ or directly with its tangent and normal cones as in nonlinear optimization.

(I'm focusing on optimal control of PDEs here; the situation may be different in optimal control of ODEs, where (non-smooth) Pontryagin-type maximum principles seem to be more popular.)

In general, whether you derive necessary optimality conditions using constraint qualifications, (rigorous) Lagrangian approaches, generalized chain rules, or directly using adjoint equations is also a matter of taste -- in the end, you should end up with the same system, and the crucial condition that you need to verify is also the same at heart: the surjectivity of $F'(\bar x)^*$ is needed to verify the Zowe--Kurcyusz condition, the existence of a Lagrange multiplier, the applicability of the chain rule, or the solvability of the adjoint equation.

However, while an abstract, "one-size-fits-all", theory is nice, it can get to the point where verifying the abstract conditions can be more work than deriving an optimality system by hand, in particular since the spaces only match up nicely in simple examples. (For example, in the initial example, $Y$ needs to be a subspace of the space of continuous functions, while your solution operator will map to some Sobolev space; this means you need embedding theorems that restrict the space dimension you can consider.) On the other hand, showing existence of a solution to a specific adjoint equation can make use of the specific right-hand side etc. In addition, you can make use of additional properties such as hidden regularity of the adjoint equation to derive additional properties of the control that you don't see in the abstract formulation.

The situation gets even worse in the non-smooth case: While you can derive abstract optimality conditions using generalized tangent and normal cones in Banach spaces [3,4], if you try to actually characterize them for concrete optimal control problems so you can get explicit optimality conditions, you may find these to be -- in the words of a colleague -- "unpleasantly large" [5]. In these cases, you may actually get stronger -- or at least more explicit -- conditions by deriving them by hand (e.g., through a limiting process on the level of the PDE).

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^{[1] Tröltzsch, Fredi, Optimal control of partial differential equations. Theory, procedures, and applications., Wiesbaden: Vieweg (ISBN 3-528-03224-3). x, 297 p. (2005). ZBL1142.49001.}

^{[2] Zowe, J.; Kurcyusz, S., Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optimization 5, 49-62 (1979). ZBL0401.90104.}

^{[3] Mordukhovich, Boris S., Variational analysis and applications, Springer Monographs in Mathematics. Cham: Springer (ISBN 978-3-319-92773-2/hbk; 978-3-319-92775-6/ebook). xix, 622 p. (2018). ZBL1402.49003.}

^{[4] Ioffe, Alexander D., Variational analysis of regular mappings. Theory and applications, Springer Monographs in Mathematics. Cham: Springer (ISBN 978-3-319-64276-5/hbk; 978-3-319-64277-2/ebook). xxi, 495 p. (2017). ZBL1381.49001.}

^{[5] Harder, Felix; Wachsmuth, Gerd, The limiting normal cone of a complementarity set in Sobolev spaces, Optimization 67, No. 10, 1579-1603 (2018). ZBL1407.49010.}