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


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.

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Appendix: Proof of Theorem 3.1

Appendix: Proof of Theorem 3.1

Consider the following system:

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

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 gH1(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 \(C:=C(M, {\Omega }, a_{0}, a_{1}, \overline {y})\)


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 (wj) of the spectral problem ((wj,v)) = λj(wj,v), for all \(v\in V=H^{3}({\Omega })\cap {H^{1}_{0}}({\Omega })\) and j = 1, 2, 3,... We represent by Vm the subspace of V generated by vectors {w1,w2,...,wm}. We propose the following approximate problem:

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

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 zm in the usual ways:

Estimate I::

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

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


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

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 = −Δzm(t) in Eq. A.67, we see that

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

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

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

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

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

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

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

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

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

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

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

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

Therefore, we can confirm that

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

We argue by contradiction and using Eqs. A.73A.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 {zm} 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

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

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

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


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

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 : ZZ given by \({\Lambda } (\hat {p}^{1},\hat {p}^{2})=(\tilde {p}^{1},\tilde {p}^{2})\) is compact and Λ(BK[0,R0]) ⊂ BK[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.

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Huaman, D.N. Stackelberg-Nash Controllability for a Quasi-linear Parabolic Equation in Dimension 1D, 2D, or 3D. J Dyn Control Syst (2021).

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  • Parabolic nonlinear PDEs
  • Controllability
  • Stackelberg-Nash strategies
  • Carleman inequalities

Mathematics Subject Classification (2010)

  • 35B37
  • 93C20
  • 93B05