Pedro Blöss Braga
1 Introduction
We aim to develop a comprehensive theory of optimal control for nonlinear parabolic equations of Leray-Lions type, whose solutions may fail to exist in the classical weak sense for low regularity data.
In this case, the appropriate notion of solution is the renormalized solution, introduced by Lions and Di Perna [8] for the study of the Boltzmann equation, and later extended to nonlinear elliptic and parabolic equations by Blanchard et al [2–4, 6, 9–11].
In this work, we investigate the optimal control problem where the control f belongs to L^p for 1 \leq p \lt \infty, and we are particularly interested in the case of forcing terms in L^1, where the control space is not reflexive, and the
lack of regularity prevents the use of the solution itself as a test function in the weak formulation.
We will study the existence of optimal controls, applying the direct method of calculus of variations, and a \Gamma-convergence analysis, given that we obtain a family of truncated costs for the optimal control problem.
Now, we show a general PDE we use for analysis, with a setting similar to [5].
1.1. Nonlinear Evolution Equations
Given a bounded domain \Omega \subset \mathbb{R}^N, N \geq 1, T>0, Q = (0,T) \times \Omega, and real numbers 1 \lt q \lt \infty and q' = \frac{q}{q-1}, consider the nonlinear evolution equation problem
\begin{aligned} \partial_t u - \div \big( a(t,x,u,Du) \big) + \div (\phi(u)) & = f & \text{ in } (0,T)\times \Omega \\ u& =0 & \text{ on } (0,T)\times \partial\Omega \\ u(t=0) & = u_0 & \text{ in } \Omega . \qquad (1.1) \end{aligned}
The operator -\div \big( a(t,x,u, Du) \big) is a Leray-Lions operator, which is coercive, and grows like |Du|^{q-1} with respect to Du (see assumption (1.8)).
Also, \phi:\mathbb{R} \to \mathbb{R}^N is assumed to be continuous, f \in L^p(Q), and u_0 \in L^p(\Omega).
As by [5], since the solution itself is not an admissible test function for (1.1), truncation functions are introduced to the problem, providing bounded functions with sufficient regularity.
There are several applications derived from (1.1), such as the p-Laplace problem (nonlinear conduction), flow in porous media, non-Newtonian fluid flows, image processing, et cetera.
Definition 1.1. (Truncation function)
We denote by T_K the truncation function at height K \geq 0 for argument r as \displaystyle T_K(r) = \min \big( K, \max(r, -K) \big).
Next, we show a general definition of renormalized solutions for (1.1), which overall rely on
testing the equation with bounded functions obtained through truncations of the solution.
1.2 Renormalized solutions
Definition 1.2.
A function u \in L^{\infty}(0, T; L^p(\Omega)) is a renormalized solution of (1.1), if
T_K(u) \in L^q(0, T; W^{1,q}_0(\Omega)), \quad \forall K \geq 0, \qquad (1.2)
\int_{\{ (t,x) \in Q; n \leq |u(t,x)| \leq n + 1 \}} a(t,x, u,Du) \cdot Du \, dx dt \to 0 \text{ as } n \to \infty, \qquad (1.3)
for every S \in W^{2, \infty}(\mathbb{R}) piecewise C^1 and such that S' has compact support,
\partial_t S(u) - \div( S'(u) a(u, Du)) + S''(u) a(u,Du)Du + \div( S'(u) \phi(u)) \\ - S''(u)\phi(u) Du = f S'(u) \qquad (1.4)
S(u)(t=0) = S(u_0) \text{ in } \Omega . \qquad (1.5)
The additional regularity u \in L^{\infty}(0, T; L^p(\Omega)) can be obtained through a priori estimates on approximate solutions of (1.1), using p-weighted truncated test functions and the assumption u_0 \in L^p(\Omega).
1.3 Optimal Control Problem
We consider the following optimal control problem
\begin{aligned} & \underset{ f \in \mathcal{F}_{ad}}{\text{minimize}} & & J(u,f) := \underbrace{ \frac{\gamma}{p} \| u - u_d\|^p_{L^p\big( 0,T; L^p(\Omega) \big)} }_{\text{tracking}} + \underbrace{ \frac{\alpha}{p} \| f\|^p_{L^p\big( 0,T; L^p(\Omega) \big)} }_{\text{regularization}} \\ & \text{subject to} & & \partial_t u - \div (a(t,x,u,Du)) + \div (\phi(u)) = f \quad \text{ in } (0,T)\times \Omega \\ &&& u =0 \quad \quad \quad \quad \,\, \text{ on } (0,T)\times \partial\Omega , \\ &&& u(t=0) = u_0 \quad \text{ in } \Omega , \\ &&& 0 \leq f \leq f_{\text{max}} \quad \text{ in } (0, T) \times \Omega . \end{aligned} \qquad (OCP)
Next, we show the assumptions taken, which are required for the well-posedness of renormalized solutions.
Definition 1.3 (Assumptions)
Following [5], and the domain defined in the PDE problem (1.1), we assume:
a:Q \times \mathbb{R} \times \mathbb{R}^N \to \mathbb{R}^N \text{ is a Carathéodory function,} \qquad (1.6)
a(t,x,s,\xi) \xi \geq \alpha |\xi|^q \text{ for a.e. } (t,x) \in Q, \forall s \in \mathbb{R}, \forall \xi \in \mathbb{R}^N, \alpha \in \mathbb{R}^+, \qquad (1.7)
\text{ For any } K>0, \text{ there exists a } \beta_K>0 \text{ and a }C_K \in L^{q'}(Q) \text{ such that } \\ |a(t,x,s,\xi)| \leq C_K(t,x) + \beta_K|\xi|^{q-1} , \text{ for a.e. } (t,x) \in Q, \forall s: |s| \leq K, \text{ and } \forall \xi \in \mathbb{R}^N, \qquad (1.8)
\big( a(t,x,s,\xi) - a(t,x,s,\xi')\big) \big( \xi - \xi' \big) \geq 0, \\ \forall s \in \mathbb{R}, \forall \xi, \xi'\in \mathbb{R}, \text{ and for a.e. } (t,x) \in Q, \qquad (1.9)
\phi: \mathbb{R} \to \mathbb{R}^N \text{ is a continuous function}, \qquad (1.10)
f \in L^p(Q), \qquad (1.11)
u_0 \in L^p(\Omega) . \qquad (1.12)
These assumptions are required for the existence and uniqueness of renormalized solutions for (1.1). For further details on existence and uniqueness refer to [1, 5]
In the stability results for well-posedness of renormalized solutions for nonlinear evolution equations, [5] shows that under some assumptions on the PDE functions and weak convergence of controls in L^p(Q), it is possible to obtain convergence of truncated sequences in a certain Bochner space. This result may be very useful for our analysis when using embeddings to obtain convergence of truncations in L^p(Q), and it is quickly introduced next.
2 Important properties of renormalized solutions
A key stability property of renormalized solutions is the weak convergence of truncations.
More precisely, [5] shows that if a is monotone, f_n \rightharpoonup f in L^1(Q) and u_{0,n} \to u_0 in L^1(\Omega), then for any K>0
T_K(u_n) \rightharpoonup T_K(u) \text{ in } L^q(0,T;W^{1,q}_0(\Omega)) .\qquad (2.1)
If the operator a is strictly monotone, the convergence (2.1) is strong.
This result can be generalized for f \in L^p(Q), 1 \lt p \lt \infty, and by Sobolev embeddings the convergence (2.1) also holds in L^p \bigl(0,T; L^p(\Omega) \bigr).
This stability result is critical when passing to the limit in the sequence of controls in the proof of existence for optimal controls.
In the case of p=1, since L^1 is not reflexive, weak convergence of controls can be obtained through the Dunford-Pettis Theorem under an additional assumption of equi-integrability (see Theorem 4.30 in [7]).
Another useful property of truncations is the strong convergence
T_K(u) \to u \text{ in } L^p(Q) \text{ as } K \to \infty ,
which follows from the dominated convergence theorem.
These properties will be used when passing to the limit in minimizing sequences and in the proof of the \Gamma-convergence of truncated optimal control problems.
3 Existence of Optimal Controls
To state the existence of optimal controls for (OCP), one can apply the direct method of calculus of variations, showing that for a fixed truncation level K>0, there exists an optimal control f_K^*, for a truncated version of the cost, namely
J \bigl(T_K(u(f^*_K), f^*_K \bigr) = \frac{\gamma}{p} \| T_K \bigl( u (f^*_K) \bigr) - T_K(u_d)\|^p_{L^p\big( 0,T; L^p(\Omega) \big)} + \frac{\alpha}{p} \| f^*_K \|^p_{L^p\big( 0,T; L^p(\Omega) \big)} .
Note that the subscript K denotes the fixed choice of K>0 a priori, and that the tracking term is truncated, and we can also use the simplified notation J_K(f^*) := J \bigl(T_K(u(f^*_K), f^*_K \bigr).
Then, for a family of truncated costs J_K(f^*), in the limit of K \to \infty one can show \Gamma-convergence, i.e., J_K(f^*) \overset{\Gamma}{\to}J(f^*). Overall, this proof consists of showing the inequalities
\limsup_{K \to \infty} J_K(f^*) \leq J(f^*) \leq \liminf_{K \to \infty} J_K(f^*) .
Thus, it can be shown that the control f^* is optimal for (OCP).
4 Further Research
For (OCP) to be well-defined, a priori estimates on the PDE (1.1) applied on approximated solutions and with a p-weighted truncated test function enable u \in L^{\infty} \bigl( 0,T; L^p(\Omega) \bigr).
Moreover, one could study if some assumptions could potentially be weakened, such as replacing the strict monotonicity of a by general monotonicity, resulting in a weak convergence of truncations.
First order conditions should follow by computing the directional derivatives of the Lagrangian of (OCP), after establishing well-posedness and differentiability of the control-to-state map S: f \mapsto u(f).
After computing the second-order optimality conditions, it could also be interesting to run numerical simulations of the problem.
References
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