Stackelberg-Nash Controllability for a Quasi-linear Parabolic Equation in Dimension 1D, 2D, or 3D

Abstract

This paper deals with the application of Stackelberg-Nash strategies to the control to quasi-linear parabolic equations in dimensions 1D, 2D, or 3D. We consider two followers, intended to solve a Nash multi-objective equilibrium; and one leader satisfying the controllability to the trajectories.

Appendix: Proof of Theorem 3.1

Consider the following system:

$$ \left\{ \begin{array}{lll} z_{t} - \nabla \cdot (a(z+\overline{y}) \nabla z) -\nabla\cdot B(z)= g& \text{in} &Q,\\ \noalign{}\phantom{} z(x,t) = 0& \text{on}&{\Sigma},\\ \noalign{}\phantom{} z(x,0) = z^{0}(x)\ & \text{in}& {\Omega}. \end{array} \right. $$

(A.66)

In order to prove the existence of solution, we will first study the existence of solution for Eq. A.66. We have the following:

Lemma A.1

There exists r > 0 such that for each $z^{0}\in H^{3}({\Omega })\cap {H^{1}_{0}}({\Omega })$ and g ∈ H¹(0,T; L²(Ω)) with ∇g(0) ∈ L²(Ω) satisfying

$$ ||z^{0}||_{H^{3}({\Omega})}+||g||_{H^{1}(0,T;L^{2}({\Omega}))}+||\nabla g(0)||\leq r, $$

the problem Eq. A.66has a unique solution z satisfying

$$ ||z||_{L^{2}(0,T;H^{2}({\Omega}))}+||z_{t}||_{L^{2}(Q)}\leq C \left( ||z^{0}||_{H^{3}({\Omega})}+||g||_{H^{1}(0,T;L^{2}({\Omega}))}+||\nabla g(0)|| \right) $$

Where $C:=C(M, {\Omega }, a_{0}, a_{1}, \overline {y})$

Proof

The proof in this lemma is obtained by the argument similar to Theorem 3 in [18]. We employ Galerkin method with the Hilbertian basis from ${H^{1}_{0}}({\Omega })$, given by eigenvectors (w_j) of the spectral problem ((w_j,v)) = λ_j(w_j,v), for all $v\in V=H^{3}({\Omega })\cap {H^{1}_{0}}({\Omega })$ and j = 1, 2, 3,... We represent by V_m the subspace of V generated by vectors {w₁,w₂,...,w_m}. We propose the following approximate problem:

$$ \left\{\begin{array}{l} (z^{\prime}_{m},v)+(a(z_{m}+\overline{y})\nabla z_{m},\nabla v)+(B(z_{m}),\nabla v)= (g,v) \forall v\in V_{m}\\ \noalign{} z_{m}(0)=z_{0m}\to z^{0} \text{in} H^{3}({\Omega})\cap {H^{1}_{0}}({\Omega}) \end{array}\right. $$

(A.67)

The existence and uniqueness of (local in time) solution to the Eq. A.67 are ensured by classical ODE theory. The following estimates show that, in fact, they are defined for all t. We can get uniform estimates of the z_m in the usual ways:

Estimate I::

Taking v = z_m(t) in Eq. A.67, we deduce that

$$ \frac{1}{2}\frac{d}{dt}||z_{m}||^{2}+\frac{a_{0}}{2}{\int}_{\Omega}|\nabla z_{m}|^{2} dx\leq \tilde{C}_{1}||z_{m}||^{2}+||g||^{2} $$

(A.68)

and

$$ ||z_{m}||^{2}_{L^{\infty}(0,T;L^{2}({\Omega}))}+||\nabla z_{m}||^{2}_{L^{2}(Q)}\leq \tilde{C}_{2}(||z^{0}||^{2}+||g||^{2}_{L^{2}(Q)}) $$

(A.69)

In the sequel, the symbol $\tilde {C}_{k}$ is a constant that only depends on $M, {\Omega }, a_{0}, a_{1}, \overline {y}$, for k = 1, 2,..., 20

Estimate II::

Taking v = −Δz_m(t) in Eq. A.67, we see that

$$ \frac{1}{2}\frac{d}{dt}||\nabla z_{m}||^{2}+\frac{a_{0}}{4}||{\Delta} z_{m}||^{2}\leq \tilde{C}_{2}||\nabla z_{m}||^{2}+ \tilde{C}_{3}||{\Delta} z_{m}||^{4}+\frac{2}{a_{0}\tilde{C}_{11}}||g||^{2} $$

(A.70)

Estimate III::

Taking $v=-{\Delta } z^{\prime }_{m}$, we deduce that

$$ ||\nabla z^{\prime}_{m}||^{2}+\frac{1}{2}\frac{d}{dt}\left( {\int}_{\Omega}a(z_{m}+\overline{y})|{\Delta} z_{m}|^{2} dx\right)\leq \tilde{C}_{4}||{\Delta} z_{m}||^{2} + \frac{a_{0}}{8\tilde{C}_{8}}||{\Delta} z^{\prime}_{m}||^{2}+\frac{8}{a_{0}\tilde{C}_{8}}||g||^{2} $$

(A.71)

Estimate IV::

Taking derivative in the Eq. A.67₁ with respect t and taking $v=-{\Delta } z^{\prime }_{m}$, we have

$$ \begin{array}{@{}rcl@{}} &&\ \frac{1}{2}\frac{d}{dt}||\nabla z^{\prime}_{m}||^{2}+\frac{a_{0}}{2}||{\Delta} z^{\prime}_{m}||^{2}\leq \tilde{C}_{5}||{\Delta} z_{m}||^{4}+\tilde{C}_{6}||{\Delta} z_{m}||^{2} ||{\Delta} z^{\prime}_{m}||^{2}\\ &&\qquad\qquad +\tilde{C}_{7}||{\Delta} z_{m}||^{2}+\tilde{C}_{8}||\nabla z^{\prime}_{m}||^{2}+\tilde{C}_{9}(||g||^{2}+||g^{\prime}||^{2}) \end{array} $$

(A.72)

We will denote by $\tilde {C}_{10}=\tilde {C}_{9}+\frac {8}{a_{0}}$, $\tilde {C}_{11}=2\tilde {C}_{4}\tilde {C}_{8}+\tilde {C}_{7}$, $\tilde {C}_{12}=\frac {8\tilde {C}_{11}}{a_{0}}$ and $\tilde {C}_{14}=\tilde {C}_{2} \tilde {C}_{12}$. Now, multiplying Eq. A.71 by $2\tilde {C}_{8}$, Eq. A.70 by $\tilde {C}_{12}$, Eq. A.68 by $\tilde {C}_{15}$ and adding these terms, from Eq. A.69, we have

$$ \begin{array}{@{}rcl@{}} \frac{1}{2}\frac{d}{dt}\left( \tilde{C}_{15}||z_{m}||^{2}+\tilde{C}_{12}||\nabla z_{m}||^{2}+\tilde{C}_{8}{\int}_{\Omega}a(z_{m}+\overline{y})|{\Delta} z_{m}|^{2} dx+||\nabla z^{\prime}_{m}||^{2}\right)\\ \noalign{}\phantom{aaa} + \tilde{C}_{14}||\nabla z_{m}||^{2}+ (\tilde{C}_{11}-\tilde{C}_{16}||{\Delta} z_{m}||^{2})||{\Delta} z_{m}||^{2}+\left( \frac{a_{0}}{4}-\tilde{C}_{17}||{\Delta} z_{m}||^{2}\right)||{\Delta} z^{\prime}_{m}||^{2}\\ \noalign{}\phantom{QQQQQQ} \leq \tilde{C}_{19}\left( ||z^{0}||^{2}+||g||^{2}_{L^{2}(Q)}\right)+\tilde{C}_{20}\left( ||g||^{2}+||g^{\prime}||^{2}\right)\\ \end{array} $$

(A.73)

From Eq. A.67 taking $v=-{\Delta } z^{\prime }_{m}(0)$, we have

$$ ||\nabla z^{\prime}_{m}(0)||^{2}\leq \tilde{C}^{2}_{21} \left( || {\Delta} z^{0}||^{2}+||z^{0}||_{H^{3}}+||\nabla g(0)||\right)^{2} $$

(A.74)

There exists 𝜖₀ > 0 such that for

$$ \left( ||z^{0}||_{H^{3}({\Omega})}+||g||_{L^{2}(Q)}+||g^{\prime}||_{L^{2}(Q)}+||\nabla g(0)||\right)<\epsilon_{0} $$

we have

$$ \left\{\begin{array}{l} \noalign{}\phantom{QQQQQ|} ||{\Delta} z^{0}||^{2}<\frac{\tilde{a}}{\tilde{C}_{16}+\tilde{C}_{17}},\\ \noalign{}\phantom{|} \frac{1}{2}\left( \tilde{C}_{15}||z^{0}||^{2}+\tilde{C}_{12}||\nabla z^{0}||^{2}+\tilde{C}_{8}a_{1}||{\Delta} z^{0}||^{2}\right)+\tilde{C}_{20}(||g||^{2}_{L^{2}(Q)}+||g^{\prime}||^{2}_{L^{2}(Q)})\\ \noalign{}\phantom{|} +\frac{1}{2}\tilde{C}^{2}_{21}\left( ||z^{0}||_{H^{3}}+||{\Delta} z^{0}||^{2}+||\nabla g(0)||\right)^{2} +\tilde{C}_{19}T\left( ||z^{0}||^{2}+||g||^{2}_{L^{2}(Q)}\right)\\ \noalign{}\phantom{WWW} <\frac{\tilde{a}\tilde{C}_{8}a_{0}}{4\left( \tilde{C}_{16}+\tilde{C}_{17}\right)}, \end{array}\right. $$

(A.75)

where $\tilde {a}=min\left \{\frac {a_{0}}{4},\tilde {C}_{11}\right \}$.

Therefore, we can confirm that

$$ ||{\Delta} z_{m}||^{2}<\frac{\tilde{a}}{\tilde{C}_{16}+\tilde{C}_{17}}, \forall t\geq 0 $$

(A.76)

We argue by contradiction and using Eqs. A.73, A.74, and A.75 . Since, the inequality Eq. A.76 is valid, we obtain that

$$ \begin{array}{@{}rcl@{}} &&(z_{m}) \text{is bounded in} L^{\infty}(0,T;{H^{1}_{0}}({\Omega})\cap H^{2}({\Omega})),\\ &&(z^{\prime}_{m}) \text{is bounded in} L^{2}(0,T;{H^{1}_{0}}({\Omega})\cap H^{2}({\Omega}))\cap L^{\infty}(0,T;{H^{1}_{0}}({\Omega})) \end{array} $$

These uniform bounds allow taking limits in Eq. A.67 (at least for a subsequence) as $m\to \infty$. Indeed, the unique delicate point is the a.e. convergence of $a(z_{m}+\overline {y})$. But this is a consequence of the fact that the sequence {z_m} is pre-compact in $L^{2}(0,T;{H^{1}_{0}}({\Omega }))$ and $a\in C^{3}(\mathbb {R})$.

The uniqueness of the strong solution to Eq. A.67 can be proved by argument standards (to see [18]). □

Proof of Theorem 3.1

Let r given by Lemma A.1 and assume that

$$ ||z^{0}||_{H^{3}({\Omega})\cap {H^{1}_{0}}({\Omega})}+||f||_{H^{1}(0,T;L^{2}(\mathcal{O}))}+||\nabla f(0)||\leq \frac{r}{2}, $$

□

we will use Schauder’s Fixed Point Theorem. Indeed, let R₀ be a constant to be determined later, let us introduce the Banach space K₁ and K₂, with

$$ \begin{array}{@{}rcl@{}} &&K_{1}=\{w\in H^{1}(0,T;H^{2}({\Omega}); \nabla w(0)\in L^{2}({\Omega})\}\\ &&K_{2}=\{w\in L^{\infty}(0,T;H^{2}({\Omega})\cap {H^{1}_{0}}({\Omega})), w_{t}\in L^{2}(0,T;{H^{1}_{0}}({\Omega}))\cap L^{\infty}(0,T;L^{2}({\Omega})),\\ &&\phantom{WWWWWW} w_{tt}\in L^{2}(0,T;H^{-1}({\Omega}))\} \end{array} $$

denote by Z = K₁ × K₁, K = K₂ × K₂ and fix $(\hat {p}^{1},\hat {p}^{2})\in B_{K}[0,R_{0}]$. Notice that

$$ \begin{array}{@{}rcl@{}} && ||z^{0}||_{H^{3}({\Omega})}+||f 1_{\mathcal{O}}-\frac{1}{\mu_{1}}\hat{p}^{1} 1_{\mathcal{O}_{1}}-\frac{1}{\mu_{2}}\hat{p}^{2} 1_{\mathcal{O}_{2}} ||_{H^{1}(0,T;L^{2}({\Omega}))}\\ &&\phantom{QQ} +||\nabla \left( f1_{\mathcal{O}}-\frac{1}{\mu_{1}}\hat{p}^{1}1_{\mathcal{O}_{1}}-\frac{1}{\mu_{2}}\hat{p}^{2}1_{\mathcal{O}_{2}}\right)(0)||\leq \frac{r}{2}+R_{0}\left( \frac{1}{\mu_{1}}+\frac{1}{\mu_{2}}\right). \end{array} $$

If we take $R_{0} = \frac {r}{2}\left (\frac {1}{\mu _{1}}+\frac {1}{\mu _{2}}\right )^{-1}$ then, by Lemma A.1, there exists $\hat {z}$ the unique solution of Eq. A.66 with $g=f 1_{\mathcal {O}}-\frac {1}{\mu _{1}}\hat {p}^{1} 1_{\mathcal {O}_{1}}-\frac {1}{\mu _{2}}\hat {p}^{2} 1_{\mathcal {O}_{2}}, \hat {z}(0)=z^{0}$ and from Eq. A.73 and Eq. A.76, we have

$$ ||\hat{z}||_{K_{2}}\leq C_{1} r $$

(A.77)

where $C_{1} := C_{1}({\Omega }, \mathcal {O}, \mathcal {O}_{i}, M, a_{0}, a_{1})$.

Now, consider $(\tilde {p}^{1},\tilde {p}^{2})\in Z$ solution of

$$ \left\{\begin{array}{lll} -\tilde{p}^{i}_{t} - a(\hat{z}+\overline{y}) {\Delta} \tilde{p}^{i} = \alpha_{i} (\hat{z}-z_{i,d}) 1_{\mathcal{O}_{id}}& \text{in}& Q,\\ \tilde{p}^{i}(x,t)=0 & \text{on} & {\Sigma},\\ \tilde{p}^{i}(x,T)=0 & \text{in} & {\Omega}, \end{array}\right. $$

satisfying

$$ \begin{array}{@{}rcl@{}} ||\tilde{p}^{i}||_{K_{2}}& \leq & C_{2} \left( ||\hat{z}||_{H^{1}(0,T;L^{2}(\mathcal{O}))}+||z_{id}||_{H^{1}(0,T;L^{2}({\Omega}))} \right)\\ \noalign{}\phantom{} & \leq & C_{2}\left( C_{1} r+||z_{id}||_{H^{1}(0,T;L^{2}({\Omega}))} \right)\\ \noalign{}\phantom{} & \leq &C_{3}, \end{array} $$

(A.78)

where $C_{3}:= C_{3}({\Omega },\mathcal {O}, \mathcal {O}_{i}, M, ||z_{id}||_{H^{1}(0,T;L^{2}({\Omega }))})$.

In this way, if μ₁ and μ₂ are sufficiently large, such that

$$C_{3} \leq \frac{r}{2}\left( \frac{1}{\mu_{1}}+\frac{1}{\mu_{2}}\right)^{-1} =R_{0}, $$

then, Λ : Z → Z given by ${\Lambda } (\hat {p}^{1},\hat {p}^{2})=(\tilde {p}^{1},\tilde {p}^{2})$ is compact and Λ(B_K[0,R₀]) ⊂ B_K[0,R₀]. By Schauder’s Fixed Point Theorem Λ has a fixed point $(\hat {p}^{1},\hat {p}^{2})$ which is, together with $\hat {z}$, a solution of Eq. 3.14.

Notice that, due to Eqs. A.77 and A.78, the solution $(\hat {z},\hat {p}^{1},\hat {p}^{2})$ satisfies

$$ ||(\hat{z},\hat{p}^{1},\hat{p}^{2})||_{K_{2}\times K_{2} \times K_{2}}\leq C_{0}. $$

The uniqueness can be proved using these last inequality and standard energy estimates.