Analysis and optimal control of some quasilinear parabolic equations

In this paper, we consider optimal control problems associated with a class of quasilinear parabolic equations, where the coefficients of the elliptic part of the operator depend on the state function. We prove existence, uniqueness and regularity for the solution of the state equation. Then, we analyze the control problem. The goal is to get first and second order optimality conditions. To this aim we prove the necessary differentiability properties of the relation control-to-state and of the cost functional.


Introduction.
In this paper, we analyze the following optimal control problem (P) min α≤u(x,t)≤β

J(u)
with J(u) = 1 2 Q (y u (x, t) − y d (x, t)) 2 dx dt + ν 2 Q u 2 (x, t) dx dt, y u being the solution of the following quasilinear partial differential equation x, t, y(x, t))∇ x y] + a 0 (x, t, y(x, t)) = u in Q = Ω × (0, T ), y(x, t) = 0 on Σ = Γ × (0, T ), y(x, 0) = y 0 (x) in Ω. (1) Above Ω ⊂ R n , 1 ≤ n ≤ 3, is a bounded open set with a C 1,1 boundary Γ ([33, Definition 1.2.1.1]), T > 0, −∞ < α < β < +∞, and ν > 0 are given numbers. Two main parts are considered in the paper. The first one concerns the analysis of the state equation and the second one, the analysis of the associated control problem. Under suitable assumptions, we prove existence, uniqueness and regularity of the solution of (1). Then, in the second part, additional differentiability assumptions on the functions a and a 0 of the state equation are imposed to prove first and second order optimality conditions for (P). In this paper, the main emphasis is to establish the theoretical results needed to carry out the numerical analysis of the control problem. Indeed, the regularity of the optimal controls deduced from the first order optimality conditions, and the sufficient second order optimality conditions are the key tools to prove error estimates for the numerical approximation of the control problem. The numerical analysis will be done in a forthcoming paper.
However, just a few papers are devoted to the control of quasilinear equations of p-Laplace type: [10], [14], [15], [16], [18], [26], [41]. However, control problems where the nonlinearity is in the state, not in the gradient, have not been extensively studied. These equations are not of monotone type. In the elliptic case, we mention [12], [13], [21], [22]. We only know two papers for this type of nonlinearity in the parabolic case: [5] and [31]. In [31], the nonlinearity is quite general, but the author imposes strong assumptions on the corresponding functions, which are not required in our formulation. An important difference of our problem with the one studied in [5] is that we do not assume the boundedness of the nonlinear function a(x, t, y). Additionally, we prove more regularity for the optimal control and state, which is essential to prove error estimates for the numerical approximations.
Before proving this theorem we establish the following lemma.
To prove the existence of a solution for this problem we use the Schauder's fixed point theorem as follows. Let F : L 2 (Q) −→ L 2 (Q) be the functional associating with each element w ∈ L 2 (Q) the unique solution F (w) = y w ∈ W (0, T ) of the linear problem y(x, t) = 0 on Σ, y(x, 0) = y 0 (x) in Ω.
Finally, we prove the uniqueness. Let us assume that y 1 and y 2 are solutions of (1) belonging to C(Q) ∩ W 0,1 p,q (Q) and take y = y 2 − y 1 . Subtracting the equations satisfied by y 2 and y 1 we get Multiplying this equation by y and integrating in Ω × (0, t) we get with M = max{ y 2 ∞ , y 1 ∞ } and using the monotonicity of a 0 with respect to y From here we get With Gronwall's inequality we infer that y = 0, which proves the uniqueness.
Remark 1. Theorem 2.1 and Lemma 2.2 are still valid if we only assume Lipschitz regularity of Γ and 2 ≤ q < q * for some q * > 4 if n = 2, q * > 3 if n = 3, and arbitrarily big q * < ∞ if n = 1. Indeed, observe that the C 1,1 regularity of Γ was only used in the proof of Lemma 2.2 to claim that the operator − defines an isomorphism between W 1,q 0 (Ω) and W −1,q (Ω) for every t ∈ [0, T ], which is still certain in the range of q indicated; see [36] and [29,Lemma 6.2]. Finally, if Ω is a convex polygonal or polyhedral domain, then A(t) : W 1,q 0 (Ω) → W −1,q (Ω) is an isomorphism for every t ∈ [0, T ] and every 1 < q < ∞; [27]. Therefore, Theorem 2.1 and Lemma 2.2 hold under this assumption as well.
Some additional regularity of the solution of (1) is established.
p,q (Q). Moreover, there exists a constant M p,q depending on y 0 and u such that To prove the above theorem we will use the following lemmas.
We assume that f ∈ L p (0, T ; L q (Ω)) and y 0 ∈ B q,p with 1 < p, q < ∞. Then the equation has a unique solution y ∈ W 2,1 p,q (Q) and the following inequality holds Then, there exists a unique solution y ∈ W 2,1 p,q (Q) of (9) and its norm is estimated by the norms of ∇ x b, f and y 0 in their corresponding spaces.
3. Analysis of the control problem. In this section, besides Assumptions 1 and 2, we make the following hypotheses.

EDUARDO CASAS AND KONSTANTINOS CHRYSAFINOS
The set of admissible for the control problem (P) is denoted by It is immediate that the space of controls L ∞ (Q) is continuously embedded in U . The following theorem establishes the existence of a unique solution for every element u ∈ U .
Let us consider the mapping G : U −→ Y given by G(u) = y u solution of (1). Then, we have the following differentiability properties. and Proof. We apply the implicit function theorem. To this end we define the mapping Looking at the definition of Y and using the assumptions along with (4) it is easy to check that F is well defined. Moreover, F is obviously of class C 2 and ∂F ∂y (y, u)z Since we have that F (y u , u) = (0, 0), the theorem follows from the implicit function theorem if we prove that ∂F ∂y (y u , u) : (Ω) is an isomorphism. This is equivalent to the existence, uniqueness and continuous dependence of the (Ω) ⊂ C 0 (Ω) and the assumptions (1.1)-(1.6) and (7.1)-(7.2) of [38, Chapter 3] hold. Hence, the existence and uniqueness of a solution z ∈ L 2 (0, T ; H 1 0 (Ω)) ∩ C(Q) follows. Finally, the regularity in W 1,0 p (Q) follows from Lemma 2.2. We only need to move the term −div x ∂a ∂y (x, t, y u )∇ x y u z to the right hand side of the equation. To treat this term we observe that ∂a ∂y (x, t, y u )∇ x y u z ∈ Lp(Q) and hence div x ∂a ∂y (x, t, y u )∇ x y u z ∈ Lp(0, T ; W −1,p (Ω)).
As a consequence of theorems 3.2 and 3.3 we get the differentiability of the cost functional J.
Proof. The differentiability of J follows from Theorem 3.2 and the chain rule. The only thing that we have to prove is the existence and uniqueness of the solution of (43). From our assumptions, the regularity of y u ∈ Y established in Theorem 3.1 and the fact that y d ∈ L ∞ (Q), the existence and uniqueness of a solution ϕ u ∈ L 2 (0, T ; H 1 0 (Ω)) ∩ C(Q) follows from [38,Chap. 3]. Now we observe that the equation (43) can be written in the form ϕ(x, t) = 0 on Σ, ϕ(x, T ) = 0 in Ω. with Applying the results of [43] to this equation, we get that ϕ u ∈ H 2,1 (Q). Let us prove that prove that f ∈ L p (Q) ∀p ∈ [2, ∞). This is done by a bootstrap argument. We use the interpolation inequality of Gagliardo-Nirenberg [9, p. 313] Taking r = 2 in (33) and using that ϕ u ∈ L ∞ (Q) ∩ L 2 (0, T ; H 2 (Ω)), we deduce that ∇ x ϕ u ∈ L 4 (Q), therefore f ∈ L 4 (Q) and the regularity ϕ u ∈ L 4 (0, T ; W 2,4 (Ω)) follows from [43] again. Again, we use (33) with r = 4 to deduce that ∇ x ϕ u ∈ L 8 (Q) and, hence, f ∈ L 8 (Q) and ϕ u ∈ W 2,1 8 (Q). Repeating this argument we conclude that ϕ u ∈ W 2,1 p (Q) ∀p ∈ [2, < ∞). It is obvious that (29) defines a linear and continuous form in L 2 (Q). Let us prove that (30) is a bilinear continuous form in L 2 (Q). From Theorem 3.3 we know that z v1 , z v2 ∈ W (0, T ) and the mapping v → z v is continuous from L 2 (Q) to W (0, T ). The W 2,1 p (Q) regularity of ϕ u for p arbitrarily big implies that ϕ u ∈ L ∞ (Q). Moreover, from (18) and (19) we have that ∂a0(x,t,yu) ∂y ∈ L ∞ (Q) and ∂ 2 a0(x,t,yu) ∂y 2 ∈ L ∞ (Q). Therefore, to conclude the the proof it is enough to estimate the terms ∇ x ϕ u ∇ x z v1 v 2 and ∇ x ϕ u ∇ x y u z v1 z v2 in L 1 (Q) in terms of v 1 L 2 (Q) v 2 L 2 (Q) . This is obtained by using the generalized Hölder's inequality, and arguing as at the end of the proof of Theorem 3.3 to deduce the estimates in L 3 (Q), as follows Remark 3. The C 1,1 regularity can be relaxed for Theorems 3.1-3.4. Indeed, let us assume that Ω is a convex polygonal or polyhedral domain. Of course, the comments of Remarks 1 and 2 apply to deduce the regularity result y u ∈ W 1,0 p (Q) established in Theorem 3.1. From Remark 1 we also deduce that Theorem 3.2 is still valid. Theorem 3.3 is valid as well. We can argue in the same way, we only have to change the estimate (28). Thus, in dimension n = 3, recalling thatp > n + 2 = 5, we have ∇ x y u z L 2 (Ω) ≤ ∇ x y u L 5 (Ω) z L 10/3 (Ω) . Now using again an interpolation inequality in the L p spaces, [9, page p3], we get Combining the above inequalities we deduce ∇ x y u z L 2 (Q) ≤ C z
We finish the paper by establishing the first and second order optimality conditions. In what follows, we say thatū ∈ U ad is a local solution of (P) if there exists a ball B ε (ū) ⊂ L 2 (Q) such that As usual, we say thatū ∈ U ad is a solution of (P) if the above inequality holds for every u ∈ U ad . Theorem 3.5. (P) has at least one solution. Moreover, for any local solutionū of (P) there existȳ ∈ W 2,1 p (Q) andφ ∈ W 2,1 p (Q) ∀p < ∞ such that the following optimality system holds y(x, t) = 0 on Σ,ȳ(x, 0) = y 0 (x) in Ω.
Theorem 3.6. Ifū is local minimum of (P), then J (ū)v 2 ≥ 0 ∀v ∈ Cū. Reciprocally, ifū ∈ U ad satisfies (24) and J (ū)v 2 > 0 ∀v ∈ Cū \ {0}, then there exist δ > 0 and ε > 0 such that This theorem follows from the abstract results proved in [23]. Indeed, it is easy to check that the assumptions (A1) and (A2) of [23] hold for A = U ∞ = L ∞ (Q), K = U ad , U 2 = L 2 (Q) and Λ = ν. To check these assumptions it is enough to recall the expression of J given in (30) and to take into account that the weak convergence v k v in L 2 (Q) implies the weak convergence z v k z v in W (0, T ), hence the strong convergence z v k → z v in L 2 (Q).
These changes lead to the obvious modifications in (37) and (38). Finally, Theorem 3.6 is also valid and it follows from the abstract results proved in [23].