Numerical Stackelberg--Nash Control for the Heat Equation

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. SCI. COMPUT.©2020 Society for Industrial and Applied Mathematics Vol. 42, No. 5, pp. A2678--A2700 NUMERICAL STACKELBERG--NASH CONTROL FOR THE HEAT EQUATION\ast PIT\' AGORAS P. DE CARVALHO\dagger AND ENRIQUE FERN\' ANDEZ-CARA\ddagger \bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . This paper deals with a strategy to solve numerically control problems of the Stackelberg--Nash kind for heat equations with Dirichlet boundary conditions. We assume that we can act on the system through several controls, respecting an order and a hierarchy: a first control (the leader) is assumed to choose the policy; then, a Nash equilibrium pair, determined by the choice of the leader and corresponding to a noncooperative multiple-objective optimization strategy, is found (these are the followers). Our method relies on a formulation inspired by the work of Fursikov and Imanuvilov. More precisely, we minimize over the class of admissible null controls a functional that involves weighted integrals of the state and the control, with weights that blow up at the final time. The use of the weights is crucial to ensure the existence of the controls and the associated state in a reasonable space. We present several mixed formulations of the problems and, then, associated mixed finite element approximations that are easy to handle. In a final step, we exhibit some numerical experiments making use of the Freefem++ package. \bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs . controllability, Stackelberg--Nash strategies, heat equation \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs . 34K35, 35Q93, 49J20, 90C29 \bfD \bfO \bfI . 10.1137/19M1253320 1. Introduction. The controllability of linear and nonlinear PDEs has been the subject of a lot of work during the last few decades. Theoretical and numerical aspects and their connection to applications have been considered by many authors and a lot of advances can be mentioned. Let us mention the papers [15, 29, 25, 20, 13, 33] and [9, 24, 8, 7], respectively dealing with related theoretical and numerical analysis. In particular, it is often interesting to try to control the system acting from several sides. It is expected that this makes it possible to govern the solution in a sharp way, in the sense that not only one but several observed properties behave as desired. A possible way to do this is through a hierarchic strategy: we fix several goals, we establish an order of priority, and we choose accordingly the controls with the aim to achieve all them. This paper deals with the numerical solution of a null controllability problem for the heat equation through a hierarchy of controls. More precisely, we have chosen the so-called Stackelberg--Nash method, which can be briefly described as follows: \bullet We have control of two kinds: leaders and followers. \bullet We associate to each leader a Nash equilibrium pair (two followers) that corresponds to a noncooperative multiple-objective optimal control problem. \bullet Then, we choose the leader among the set of null controls by minimizing a suitable functional. \ast Submitted to the journal's Methods and Algorithms for Scientific Computing section March 29, 2019; accepted for publication (in revised form) July 7, 2020; published electronically September 9, 2020. https://doi.org/10.1137/19M1253320 \bfF \bfu \bfn \bfd \bfi \bfn \bfg : The work of the second author was partially supported by grant MTM2016-76990-P. \dagger Universidade Estadual do Piau\'{\i}, UESPI, Teresina, PI, 64002-150, Brazil (pitagorascarvalho@ yahoo.com.br). \ddagger Departamento EDAN, Universidad de Sevilla, 41080 Sevilla, Spain ([email protected]). A2678 Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. NUMERICAL STACKELBERG--NASH CONTROL A2679 Hence, for the resolution of the control problem our tasks are, first, to prove that we can associate to any admissible leader a Nash pair and, then, to prove that the whole system (controlled by the leader and the followers) satisfies the desired null controllability property. For instance, if we interpret that the controls are heat sources applied at different locations of a room and the state is the room temperature, it is completely meaningful to try to guide the temperature at rest at the end of the day and, additionally, keep the temperature ``not too far"" from prescribed values at some prescribed domains. It is shown in [1, 2] that this is possible to achieve; in fact, it is realizable even if the considered state equation is semilinear and under other more complicated circumstances. We can find many other situations where hierarchic control can be applied. Thus, Stackelberg--Nash techniques can be useful for traffic control problems. A typical problem is to minimize the queue in each road of an intersection, taking into account that there is a road that can enforce its strategy on the others; see [26]. Other fields where these techniques can be of help are finance, production and marketing, economics of growth, etc.; see, for instance, [30, 31]. For other applications, see [5]. At this point, it is important to mention that the numerical solution of controllability problems for PDEs is not, in general, a simple task. A lot of authors have contributed to this effort but it has become clear that straightforward reduction to finite dimensions can lead to ill-posedness and is not necessarily a good idea; see [9, 7, 32] for some explanations. We will follow here a strategy relying on the well-known Fursikov--Imanuvilov formulation of the null controllability of linear evolution PDEs; see [20]. This has also been the basis of the numerical controllability results in [17], [18], [19]. More precisely, we minimize over the class of admissible null controls a functional that involves weighted integrals of the state and the control, with some weights that blow up at the final time. The use of the weights is crucial to ensure well-posedness and existence of controls and associated states in reasonable spaces. We present several mixed formulations of the problems and, then, associated mixed finite element approximations that are easy to handle. At the end of the paper, we exhibit some numerical experiments. The computations have been carried out with the help of the Freefem++ package. For the visualization of the results, we have used suitable MATLAB graphic tools. The plan of the paper is the following. Section 2 describes the problem formulation and its motivation. In section 3, we present the main ideas of our numerical approach. In particular, we see that the task reduces to the solution of a boundary-value problem that is fourth-order in space and second-order in time. In section 4, we present a mixed formulation and an associated numerical approximation, where we avoid the use of C1 finite elements. The methods are illustrated with several numerical experiments in section 5. Finally, section 6 contains some additional comments. 2. The problem and the motivation. Let \Omega \subset RNbe a nonempty bounded connected open set whose boundary is regular enough. Let T > 0 be given and let us consider the cylindrical domain Q:= \Omega \times (0, T ), with lateral boundary \Sigma := \partial \Omega \times (0, T). In what follows, we will denote by Ca generic positive constant. Sometimes, we will indicate the data on which Cdepends by writing C(\Omega ), C(\Omega , T), etc. The usual norm and scalar product in L2(\Omega ) will be respectively denoted by \| \cdot \| and (\cdot ,\cdot ); on the other hand, \langle \cdot ,\cdot \rangle will stand for various duality products. Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. A2680 PIT\' AGORAS P. DE CARVALHO AND ENRIQUE FERN\' ANDEZ-CARA We will consider controlled systems of the form \left\{ yt - \Delta y=f1\scrO +v11\scrO 1+v21\scrO 2in Q, y= 0 on \Sigma , y(\cdot ,0) = y0in \Omega . (2.1) In (2.1), \scrO ,\scrO 1, and \scrO 2are nonempty open subsets of \Omega ; \scrO \subset \Omega is the main control domain and \scrO 1and \scrO 2are secondary control domains (all of them are supposed to be small); 1\scrO ,1\scrO 1, and 1\scrO 2are the corresponding characteristic functions and the associated controls are f,v1, and v2. We will use the notation \scrH i:= L2(\scrO i\times (0, T )), \scrH := \scrH 1\times \scrH 2, and \scrU := L2(\scrO \times (0, T )). Let \scrO d\subset \Omega be another nonempty open set, representing an observation domain and assume that the \mu i>0 and the functions yi,d =yi,d(x, t) and the weights \rho = \rho (x, t) and \rho 0=\rho 0(x, t) are given. The Stackelberg--Nash strategy for the null controllability and biobjective optimal control problem is as follows: \bullet Let us consider the secondary cost functionals Ji(f;v1, v2) := 1 2\int \int \scrO d\times (0,T ) | y - yi,d| 2dx dt +\mu i 2\int \int \scrO i\times (0,T ) | vi| 2dx dt, i = 1,2, (2.2) where, for each (f, v2, v2)\in \scrU \times \scrH 1\times \scrH 2,yis the unique solution to (2.1). Then, for each ``leader"" f\in \scrU , we search for a Nash equilibrium (v1(f), v2(f)) \in \scrH for J1and J2. The secondary controls v1(f) and v2(f) will then be called the ``followers"" associated to f. \bullet We will also introduce the main cost functional J(f) := 1 2\int \int Q \rho 2| y| 2dx dt +1 2\int \int \scrO \times (0,T ) \rho 2 0| f| 2dx dt,(2.3) where this time for each f\in \scrU ,yis the unique solution to (2.1) with vi= vi(f). Then, we search for a minimizer fof Jin the family of controls in \scrU such that y(x, T) = 0 in \Omega .(2.4) Recall that a Nash equilibrium for J1and J2associated to fis a couple (v1(f), v2(f)) \in \scrH satisfying J1(f;v1(f), v2(f)) = min \^v1\in \scrH 1 J1(f; \^v1, v2(f)), J2(f;v1(f), v2(f)) = min \^v2\in \scrH 2 J2(f;v1(f),\^v2). (2.5) Let f\in \scrU be given. Since the Jiare C1and convex, (v1(f), v2(f)) is a Nash equilibrium if and only if \langle J\prime 1(f;v1(f), v2(f)),(\^v1,0)\rangle = 0 \forall \^v1\in \scrH 1, v1(f)\in \scrH 1,(2.6) and \langle J\prime 2(f;v1(f), v2(f)),(0,\^v2)\rangle = 0 \forall \^v2\in \scrH 2, v2(f)\in \scrH 2,(2.7) Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. NUMERICAL STACKELBERG--NASH CONTROL A2681 where J\prime istands for the derivative of Jiwith respect to (v1, v2). This can also be written in the form \int \int \scrO d\times (0,T ) (y - yd)widx dt +\mu i\int \int \scrO i\times (0,T ) vi(f)\^vidx dt = 0(2.8) \forall \^vi\in \scrH i, vi(f)\in \scrH i, i = 1,2, where wiis the derivative of ywith respect to viin the direction \^vi. Now, we will provide explicit expressions of v1(f) and v2(f) in terms of appropriate adjoint variables. Thus, let us consider the state y(the solution to (2.1) for vi=vi(f)). It is very natural to introduce the adjoint states \phi i, with \left\{ - \phi i,t - \Delta \phi i= (y - yi,d)1\scrO din Q, \phi i= 0 on \Sigma , \phi i(\cdot , T) = 0 in \Omega . (2.9) Then, using integration by parts, it is immediate that (2.8) is equivalent to \int \int \scrO i\times (0,T ) (\phi i+\mu ivi(f)) \^vidx dt = 0 \forall \^vi\in \scrH i, vi(f)\in \scrH i, i = 1,2. This directly implies that vi(f) = - 1 \mu i \phi i| \scrO i\times (0,T ), i = 1,2.(2.10) Let us gather all this information in the same system. We obtain the following: \left\{ yt - \Delta y=f1\scrO - 2 \sum i=1 1 \mu i \phi i1\scrO iin Q, - \phi i,t - \Delta \phi i= (y - yi,d)1\scrO din Q, y= 0, \phi i= 0 on \Sigma , y(\cdot ,0) = y0, \phi i(\cdot , T) = 0 in \Omega . (2.11) Recall that our goal is to get the null controllability of (2.1), with (v1, v2) being a Nash equilibrium associated to f. Therefore, our main task is to prove the existence of a control f\in \scrU such that the solution to (2.11) satisfies (2.4) and then take the vi as in (2.10). Note that the vi(f) are affine in f. However, an important issue here is that they are nonlocal functions of f. It is also adequate to enhance that, in our setting, the control problems in charge of the followers and the leader are of a different nature: the followers must solve a biobjective control problem that is completely characterized by the system satisfied by the state and the adjoints (see [23]); on the other hand, the leader must be such that the state is led to zero at final time. From well-known results in control theory, we know that a null control exists and depends continuously on the data y0for (2.11) if and only if a suitable observability inequality is satisfied by the solutions to an appropriate adjoint system. More precisely, following the ideas in [2, 1], we introduce the adjoint Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. A2682 PIT\' AGORAS P. DE CARVALHO AND ENRIQUE FERN\' ANDEZ-CARA \left\{ - \psi t - \Delta \psi = 2 \sum i=1 \gamma i1\scrO din Q, \gamma i,t - \Delta \gamma i= - 1 \mu i \psi 1\scrO iin Q, \psi = 0, \gamma i= 0 on \Sigma , \psi (\cdot , T) = \psi T, \gamma i(\cdot ,0) = 0 in \Omega , (2.12) where \psi T\in L2(\Omega ) and we note that the null controllability of (2.11) is equivalent to the existence of a constant Csuch that \| \psi (\cdot ,0)\| 2+ 2 \sum i=1 \int \int \scrO \times (0,T ) \^\rho - 2| \gamma i| 2dx dt \leq C\int \int \scrO \times (0,T ) | \psi | 2dx dt \forall \psi T\in L2(\Omega ), where \^\rho is a suitable weight. This is explained in detail in [2] and [1], where the existence of Cis established for sufficiently large \mu 1and \mu 2, provided y1,d and y2.d decay sufficiently fast to zero as t\rightarrow T; see Theorem 1 below. This paper is mainly devoted to the computation of numerical null controls for (2.11). As indicated above, the numerical solution of controllability problems is not a simple task. We will follow here a strategy relying on the well-known Fursikov-- Imanuvilov formulation of the null controllability of linear evolution PDEs; see [20]. These ideas have been applied before in [17], among other works. 3. A strategy for the computation of null control. In this section, we explain our strategy to compute a null control for (2.11). First, let us recall a controllability result from [2]. Theorem 1. There exists \beta (\Omega ,\scrO 1,\scrO 2,\scrO d, T, y1,d, y2,d)>0such that, if \mu i\geq \beta for i= 1,2, the linear system (2.11) is null-controllable at time T > 0. In other words, under the previous conditions, there exists a positive function \rho d=\rho d(x, t), blowing up as t\rightarrow T, with the following property: if \int \int \scrO d\times (0,T ) \rho 2 d| yi,d| 2dx dt < +\infty , i = 1,2,(3.1) for each y0\in L2(\Omega ) there exist controls f\in L2(\scrO \times (0, T )) and associated Nash equilibria (v1, v2)such that the corresponding solutions to (2.1) satisfy (2.4). In the reminder of this paper, we will always assume that the assumptions in Theorem 1 are satisfied. Let us introduce a formulation of the null controllability problem for (2.11) that is inspired by the results in [20]. We consider the extremal problem \left\{ Minimize K(y, \phi 1, \phi 2, f) = 1 2\Biggl( \int \int Q \rho 2| y| 2dx dt +\int \int \scrO \times (0,T ) \rho 2 0| f| 2dx dt\Biggr) Subject to (y, \phi 1, \phi 2, f)\in H(y0, T), (3.2) where the linear manifold H(y0, T) is given by H(y0, T) := \{ (y, \phi 1, \phi 2, f) : f\in \scrU ,(y, \phi 1, \phi 2, f) satisfies (2.11) and (2.4)\} . In (3.2), it is assumed that \rho and \rho 0are appropriate weight functions that blow up exponentially as t\rightarrow T. In this paper, they will be chosen as follows: Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. NUMERICAL STACKELBERG--NASH CONTROL A2683 \rho (x, t) := e\chi (x)/(T - t), \chi (x) := K1\Bigl( eK2 - e\chi 0(x)\Bigr) , \rho 0(x, t) := (T - t)3/2\rho (x, t), (3.3) where K1and K2are sufficiently large positive constants (depending on T) and \chi 0= \chi 0(x) is a regular bounded function that is positive in \Omega , vanishes on \partial \Omega , and satisfies | \nabla \chi 0| >0 in \Omega \setminus \scrO ; for a justification of the existence of \chi 0, see [20]. An immediate consequence of Theorem 1 is that (3.2) possesses exactly one solution (y, \phi 1, \phi 2, f)\in H(y0, f). Furthermore, in view of Lagrange's principle, one must have \int \int Q \rho 2yy\prime dx dt +\int \int \scrO \times (0,T ) \rho 2 0ff\prime dx dt = 0(3.4) for all (y\prime , \phi \prime 1, \phi \prime 2, f\prime ) with f\prime \in Uand \left\{ y\prime t - \Delta y\prime =f\prime 1\scrO - 1 \mu 1 \phi \prime 11\scrO 1 - 1 \mu 2 \phi \prime 21\scrO 2in Q, - \phi \prime i,t - \Delta \phi \prime i=y\prime 1\scrO din Q, y\prime = 0, \phi \prime i= 0 on \Sigma , y\prime (\cdot ,0) = 0, \phi \prime i(\cdot , T) = 0 in \Omega . (3.5) In the next lines, we are going to perform formal computations, without taking care of a rigorous justification. This way, we will arrive at an explicit formula for the solution (y, \phi 1, \phi 2, f) to (3.2). Thus, let the triplet (\psi , \gamma 1, \gamma 2) satisfy \left\{ - \psi t - \Delta \psi = 2 \sum i=1 \gamma i1\scrO d+\rho 2yin Q, \gamma i,t - \Delta \gamma i= - 1 \mu i \psi 1\scrO i+\rho 2\phi iin Q, \psi = 0, \gamma i= 0 on \Sigma , \gamma i(\cdot ,0) = 0 in \Omega (3.6) and let us introduce the notation M(y, \phi 1, \phi 2) := \Biggl( yt - \Delta y+ 2 \sum i=1 1 \mu i \phi i1\scrO i, - \phi 1,t - \Delta \phi 1 - y1\scrO 1,d , - \phi 2,t - \Delta \phi 2 - y1\scrO 2,d \Biggr) and M\ast (\psi , \gamma 1, \gamma 2) := \Biggl( - \psi t - \Delta \psi - 2 \sum i=1 \gamma i1\scrO d, \gamma 1,t - \Delta \gamma 1+1 \mu 1 \psi 1\scrO 1, \gamma 2,t - \Delta \gamma 2+1 \mu 2 \psi 1\scrO 2\Biggr) , Then, at least formally, one has M(\rho - 2M\ast (\psi , \gamma 1, \gamma 2)) = (f1\scrO ,0,0) in Q.(3.7) Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. A2684 PIT\' AGORAS P. DE CARVALHO AND ENRIQUE FERN\' ANDEZ-CARA Also, we see that \int \int Q \rho 2y\cdot y\prime dx dt =\int \int Q\Biggl( - \psi t - \Delta \psi - 2 \sum i=1 \gamma i1\scrO d\Biggr) y\prime dx dt =\int \int \scrO \times (0,T ) \psi \psi \prime dx dt for all (y\prime , \phi \prime 1, \phi \prime 2, f\prime ) as before and, consequently, taking into account (3.4), we must have \int \int \scrO \times (0,T )\bigl( \psi +\rho 2 0f\bigr) f\prime dx dt = 0 \forall f\prime \in \scrU . Thus, we deduce that (y, \phi 1, \phi 2) = \rho - 2M\ast (\psi , \gamma 1, \gamma 2), f = - \rho - 2 0\psi \bigm| \bigm| \scrO \times (0,T ),(3.8) where (\psi , \gamma 1, \gamma 2) solves (in some sense) the system \left\{ M(\rho - 2M\ast (\psi , \gamma 1, \gamma 2)) + \rho - 2 0\psi 1\scrO = 0 in Q, (\psi , \gamma 1, \gamma 2) = (0,0,0), \rho - 2M\ast \bigl( \psi , \gamma 1, \gamma 2\bigr) = (0,0,0) on \Sigma , \rho - 2M\ast (\psi , \gamma 1, \gamma 2)1(\cdot ,0) = y0, \gamma 1(\cdot ,0) = \gamma 2(\cdot ,0) = 0 in \Omega , \rho - 2M\ast (\psi , \gamma 1, \gamma 2)(\cdot , T ) = (0,0,0) in \Omega . (3.9) On the other hand, if (\psi , \gamma 1, \gamma 2) is a solution to (3.9) and (y, \phi 1, \phi 2) and fare given by (3.8), then (3.4) holds and, as a consequence, (y, \phi 1, \phi 2, f) solves (3.2). Let us now check that (3.9) is well-posed problem in an appropriate space and therefore the solution to (3.2) is indeed given by (3.8). Let us introduce the space P0:= \{ (\psi , \gamma 1, \gamma 2)\in C2(Q)3: (\psi , \gamma 1, \gamma 2) = (0,0,0) on \Sigma , \gamma 1(\cdot ,0) = \gamma 2(\cdot ,0) = 0\} (3.10) and the bilinear form m((\psi , \gamma 1, \gamma 2),(\psi \prime , \gamma \prime 1, \gamma \prime 2)) := \int \int Q \rho - 2M\ast (\psi , \gamma 1, \gamma 2)\cdot M\ast (\psi \prime , \gamma \prime 1, \gamma \prime 2)dx dt +\int \int \scrO \times (0,T ) \rho - 2 0\psi \psi \prime dx dt. Then, as a consequence of the Carleman estimates established in the following result, m(\cdot ,\cdot ) is a scalar product in P0. Proposition 1. Let us set \rho 2:= (T - t)1/2\rho and \rho 1:= (T - t) - 1/2\rho . There exists C0(\Omega ,\scrO , T)>0such that the following holds for any (\psi , \gamma 1, \gamma 2)\in P0, with h:= \gamma 1+\gamma 2: \int \int Q\biggl[ \rho - 2 2(| \psi t| 2+| \Delta \psi | 2) + \rho - 2 1| \nabla \psi | 2+\rho - 2 0| \psi | 2+\rho - 2| h| 2\biggr] dx dt \leq C0m((\psi , \gamma 1, \gamma 2),(\psi , \gamma 1, \gamma 2)). (3.11) The proof of this result can be found in [2]. Let Pbe the completion of P0 with respect to this scalar product. Then Pis a Hilbert space and the functions (\psi , \gamma 1, \gamma 2)\in Psatisfy \int \int Q \rho - 2| M\ast (\psi , \gamma 1, \gamma 2)| 2dx dt +\int \int \scrO \times (0,T ) \rho - 2 0| \psi | 2dx dt < +\infty .(3.12) Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. NUMERICAL STACKELBERG--NASH CONTROL A2685 From Proposition 1 and a standard density argument, we also have (3.11) for all (\psi , \gamma 1, \gamma 2)\in P. In particular, we see that for any (\psi , \gamma 1, \gamma 2)\in Pone has \psi \in C0([0, T - \delta ]; H1 0(\Omega )) for all \delta > 0 and, moreover, \| \psi (\cdot ,0)\| 2 H1 0\leq C m((\psi , \gamma 1, \gamma 2),(\psi , \gamma 1, \gamma 2)) \forall (\psi , \gamma 1, \gamma 2)\in P.(3.13) Let us introduce the linear form \ell :P\mapsto \rightarrow R, with \langle \ell , (\psi , \gamma 1, \gamma 2)\rangle := \int \Omega y0(x)\psi (x, 0) dx - 2 \sum i=1 \int \int \scrO d\times (0,T ) yi,d \gamma i\psi dx dt. In view of (3.13), \ell is well defined and continuous. Furthermore, from the previous considerations, the following result holds. Theorem 2. Let (y, \phi 1, \phi 2, f)be the unique solution to (3.2). Then one has (3.8), where (\psi , \gamma 1, \gamma 2)is the unique solution to the following variational equality in the Hilbert space P: \biggl\{ m((\psi , \gamma 1, \gamma 2),(\psi \prime , \gamma \prime 1, \gamma \prime 2)) = \langle \ell , (\psi \prime , \gamma \prime 1, \gamma \prime 2)\rangle \forall (\psi \prime , \gamma 1\prime , \gamma 2\prime )\in P, (\psi , \gamma 1, \gamma 2)\in P. (3.14) Obviously, (3.14) must be viewed as the weak formulation of the boundary-value problem (3.9), that is fourth-order in xand second-order in t. It is then clear that what we have to do in practice is to solve numerically (3.14). This furnishes (an approximation of) the triplet (\psi , \gamma 1, \gamma 2). Then, we use (3.8) and deduce that the controls are given by f= - \rho - 2 0\psi \bigm| \bigm| \scrO \times (0,T ), vi= - 1 \mu i \phi i\bigm| \bigm| \scrO i\times (0,T ), i = 1,2, where \phi 1and \phi 2are respectively the second and third components of \rho - 2M\ast (\psi , \gamma 1, \gamma 2). Remark 1. In realistic problems, the control variables are subject to constraints and, obviously, the previous result does not cover directly this situation. However, at least when the constraints are local in space and time and affect the followers, something can be done. Thus, let I1and I2be two nonempty closed intervals with 0\in I1\cap I2, let us take \scrH i,d =\{ v\in \scrH i:v(x, t)\in Iia.e. \} , i = 1,2,(3.15) and let us suppose that the minimization of J1and J2in (2.5) is subject to the restrictions \^v1\in \scrH 1,d and \^v2\in \scrH 2,d. Then, (v1, v2) is a related Nash equilibrium if and only if \langle J\prime 1(f;v1, v2),(\^v1 - v1,0)\rangle \geq 0,\forall \^v1\in \scrH 1,d, v1\in \scrH 1,d and \langle J\prime 2(f;v1, v2),(0,\^v2 - v2)\rangle \geq 0,\forall \^v2\in \scrH 2,d, v2\in \scrH 2,d and, arguing as in section 2, we see that this is in turn equivalent to having vi(f) = Pi\biggl( - 1 \mu i \phi i| \scrO i\times (0,T )\biggr) , i = 1,2, Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. A2686 PIT\' AGORAS P. DE CARVALHO AND ENRIQUE FERN\' ANDEZ-CARA where Pi:\scrH i\mapsto \rightarrow \scrH i,d is the orthogonal projector and \phi iis the solution to (2.9) for i= 1,2. Consequently, in this case, the hierarchic problem reduces to get the null controllability (in y) of the coupled semilinear system \left\{ yt - \Delta y=f1\scrO - 2 \sum i=1 Pi\biggl( - 1 \mu i \phi i| \scrO i\times (0,T )\biggr) in Q, - \phi i,t - \Delta \phi i= (y - yi,d)1\scrO din Q, y= 0, \phi i= 0 on \Sigma , y(\cdot ,0) = y0, \phi i(\cdot , T) = 0 in \Omega . (3.16) It can be proved that, under the assumptions in Theorem 1, there exist controls f\in \scrU and associated Nash equilibria (v1, v2)\in \scrH 1,d \times \scrH 2,d such that the corresponding states satisfy (2.4); see [1, 2] for the details. Let Phdenote a finite dimensional subspace of P. Then, a completely natural approximation of (3.14) is the following: \biggl\{ m((\psi h, \gamma 1,h, \gamma 2,h),(\psi \prime h, \gamma \prime 1,h, \gamma \prime 2,h)) = \langle \ell , (\psi \prime h, \gamma \prime 1,h, \gamma \prime 2,h)\rangle \forall (\psi \prime h, \gamma \prime 1,h, \gamma \prime 2,h)\in Ph,(\psi h, \gamma 1,h, \gamma 2,h)\in Ph. (3.17) Thus, it could seem that, in order to solve numerically the variational equality (3.14), it suffices to construct explicitly finite dimensional spaces Ph\subset P. Note, however, that this is possible but needs a lot of work and leads to expensive computations, especially in spatial dimensions N\geq 2. The reason is that, in order to get (\psi h, \gamma 1h, \gamma 2h)\in P, we need \psi h,t + \Delta \psi h\in L2 loc(Q). In practice, this means that \psi hmust possess first-order time derivatives and up to second-order spatial derivatives in L2 loc(Q). Therefore, an approximation based on a standard triangulation of Qrequires functions that must be C0in (x, t) and C1in xand this can be complicated and too expensive. Spaces of this kind are constructed, for instance, in [17, 27] for N= 1. For N\geq 2, one has to consider other spaces, based on reduced HTC,Bell, or Bogner--Fox--Schmidt finite elements; see [11, 21]. Despite its complexity, the direct approximation of (3.17) has an advantage: it is possible to adapt the standard finite element theory to this framework and deduce strong convergence results for the numerical controls and states. 4. Mixed formulations and numerical approximations of control problems of the Stackelberg--Nash kind. 4.1. A first mixed formulation. Let us introduce the Hilbert spaces L2(\rho s; Q) := \{ g:\rho sg\in L2(Q)\} (where s\in R) and the new variable Z= (z, z1, z2) := M\ast (\psi , \gamma 1, \gamma 2). Then, Z - M\ast (\psi , \gamma 1, \gamma 2) = 0 (an equality in L2(\rho - 1;Q)3). Note that this identity can also be written in the form \int \int Q (z - M\ast (\psi , \gamma 1, \gamma 2)1)\psi \prime dx dt = 0 \forall \psi \prime \in L2(\rho ;Q), \int \int Q (zi - M\ast (\psi , \gamma 1, \gamma 2)i+1)\psi \prime dx dt = 0 \forall \psi \prime \in L2(\rho ;Q), i = 1,2. Accordingly, we introduce the following mixed reformulation of (3.14): Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. NUMERICAL STACKELBERG--NASH CONTROL A2693 Fig. 3.Test 1. The computed leader control f. (a) The computed follower v1(b) The computed follower v2 Fig. 4.Test 1. The followers. Fig. 5.Test 1. The computed state. Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. A2694 PIT\' AGORAS P. DE CARVALHO AND ENRIQUE FERN\' ANDEZ-CARA Table 1 Test 1. The mesh data in the experiments. Mesh iterate Number of triangles Number of vertices 0 1586 846 1 2131 1104 2 7735 3942 3 17581 9000 (a) The L2norm of the leader (b) The L2norm of the state Fig. 6.Test 1. Evolution in time of the L2norms. Fig. 7.Test 2. The final mesh. Number of vertices: 17608. Number of elements: 9000. First, note that the numerical approximations of the resulting systems can be carried out in at least two ways: \bullet by working with finite element spaces that are C1in space, \bullet by introducing multipliers and associated mixed formulations and working with (usual) C0finite element spaces. Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. NUMERICAL STACKELBERG--NASH CONTROL A2695 Fig. 8.Test 2. The computed leader control f. (a) The computed follower v1(b) The computed follower v2 Fig. 9.Test 2. The followers. In this paper, we have chosen the second approach. In a forthcoming paper, we will be concerned with the first one. Of course, the same ideas and techniques can be applied to the computation of Stackelberg--Nash controls in many other similar situations: semilinear heat equations with (for instance) globally Lipschitz-continuous nonlinearities, noncylindrical control domains, similar boundary control problems, hierarchical null controllability for linear and semilinear wave equations, etc. Note that some theoretical results have been obtained in [3, 4]. Observe that if the state equation is nonlinear, the convexity is lost in the functionals Jiand Jand the analysis of Stackelberg--Nash controllability is much more complex. In particular, the existence of Nash equilibria is not ensured and Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. A2696 PIT\' AGORAS P. DE CARVALHO AND ENRIQUE FERN\' ANDEZ-CARA Fig. 10.Test 2. The computed state. (a) The L2norm of the leader (b) The L2norm of the state Fig. 11.Test 2. Evolution in time of the L2norms. Fig. 12.Test 3. The final mesh. Number of vertices: 17573. Number of elements: 9000. Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. NUMERICAL STACKELBERG--NASH CONTROL A2697 Fig. 13.Test 3. The computed leader control f. (a) The computed follower v1(b) The computed follower v2 Fig. 14.Test 3. The followers. Fig. 15.Test 3. The computed state. Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. A2698 PIT\' AGORAS P. DE CARVALHO AND ENRIQUE FERN\' ANDEZ-CARA (a) The L2norm of f(\cdot , t) (b) The L2norm of y(\cdot , t) Fig. 16.Test 3. Evolution in time of the L2norms. sometimes one has to work with weaker or generalized concepts. Of course, this introduces nontrivial difficulties at the numerical level. It would also be interesting to extend and/or adapt these ideas to other multiobjective control problems, such as the computation of Stackelberg--Pareto equilibria. More precisely, we can try to establish a strategy of the following kind: \bullet For each fixed f\in \scrU , find the corresponding Pareto front for (2.1), J1and J2. In other words, find the family of couples (v1, v2)\in \scrH with the following property: there exist no (v\prime 1, v\prime 2)\in \scrH such that Ji(v\prime 1, v\prime 2)\leq Ji(v1, v2) for i= 1,2, at least one of these inequalities being strict. \bullet Then, find f\in \scrU and an associated Pareto equilibrium (v1(f), v2(f)) such that (2.4) holds. It can be proved that, for each leader f, there exists a whole family of Pareto equilibria \{ (v\alpha 1(f), v\alpha 2(f)) : \alpha \in (0,1)\} . The parameter \alpha indicates how relevant one of the functionals is regarding the other one. For any fixed \alpha , it is relatively simple to prove the existence of fsuch that the state corresponding to fand (v\alpha 1(f), v\alpha 2(f)) vanishes at time t=T. Moreover, the computation of fand the associated Pareto equilibrium can be achieved with the techniques in this paper. However, it is much more difficult (and actually open) to find leader controls f such that the same holds simultaneously for a family of associated equilibria. Some theoretical and numerical results on this approach will also appear in a forthcoming work. REFERENCES [1] F.D. Araruna, E. Fern\' andez-Cara, S. Guerrero, and M.C. Santos,New results on the Stackelberg-Nash exact control of linear parabolic equations, Systems Control Lett., 104 (2017), pp. 78--85. [2] F.D. Araruna, E. Fern\' andez-Cara, and M.C. Santos,Stackelberg-Nash exact controllability for linear and semilinear parabolic equations, ESAIM Control Optim. Calc. Var., 21 (2015) pp. 835--856. Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. NUMERICAL STACKELBERG--NASH CONTROL A2699 [3] F.D. Araruna, E. Fern\' andez-Cara, and L.C. da Silva,Hierarchical exact controllability of semilinear parabolic equations with distributed and boundary controls, Commun. Contemp. Math., (2019), 1950034. [4] F.D. Araruna, E. Fern\' andez-Cara, and L.C. da Silva,Hierarchic control for the wave equation, J. Optim. Theory Appl., 178 (2018), pp. 264--288, doi:10.1007/s10957-018-1277-6. [5] D. Bauso,Game Theory with Engineering Applications, Adv. Des. Control 30, SIAM, Philadelphia, 2016. [6] D. Boffi, F. Brezzi, and M. Fortin,Mixed Finite Element Methods and Applications, Springer Ser. Comput. Math. 44, Springer, Heidelberg, 2013. [7] F. Boyer,On the penalised HUM approach and its applications to the numerical approximation of null controls for parabolic problems, in CANUM 2012, Super-Besse, ESAIM Proc., EDP Sciences, Les Ulis, 2013. [8] F. Boyer, F. Hubert, and J. Le Rousseau,Discrete Carleman estimates for elliptic operators in arbitrary dimension and applications, SIAM J. Control Optim., 48 (2010), pp. 5357--5397. [9] C. Carthel, R. Glowinski, and J.-L. Lions,On exact and approximate boundary controllability for the heat equation: A numerical approach, J. Optim. Theory Appl., 82 (1994), pp. 429--484. [10] P.P. Carvalho and E. Fern\' andez-Cara,On the computation of Nash and Pareto equilibria for some bi-objective control problems, J. Sci. Comput., 78 (2019), pp. 246--273. [11] P.G. Ciarlet,The Finite Element Methods for Elliptic Problems, Classics in Appl. Math. 40, SIAM, Philadelphia, 2002. [12] N. Cindea, E. Fern\' andez-Cara, and A. M\" unch,Numerical controllability of the wave equation through primal methods and Carleman estimates, ESAIM Control Optim. Calc. Var., 10 (2013), pp. 1076-1108. [13] J.-M. Coron,Control and Nonlinearity, Math. Surveys Monogr. 136, AMS, Providence, RI, 2007. [14] T.A. Davis,Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method, ACM Trans. Math. Software, 30 (2004), pp. 196--199. [15] H.O. Fattorini and D.L. Russell,Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Ration. Mech. Anal., 43 (1971), pp. 272--292. [16] P. Faurre,Analyse num\'erique. Notes d'optimisation, \' Ecole Polytechnique, Palaiseau, 1988. [17] E. Fern\' andez-Cara and A. M\" unch,Strong convergence approximations of null controls for 1D heat equation, S\vec{}eMA J., 61 (2013), pp. 49--78. [18] E. Fern\' andez-Cara and A. M\" unch,Numerical exact controllability of the 1-D heat equation: Duality and Carleman weights, J. Optim. Theory Appl., 163 (2014) pp. 253--285. [19] E. Fern\' andez-Cara, A. M\" unch, and D.A. Souza,On the numerical controllability of the two-dimensional heat, Stokes and Navier-Stokes equations, J. Sci. Comput., 70 (2017), pp. 819--858. [20] A.V. Fursikov and O.Y. Imanuvilov,Controllability of Evolution Equations, Lecture Notes Ser. 34, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996. [21] R. Glowinski,Finite Element Methods for Incompressible Viscous Flow, Handb. Numer. Anal. 9, Elsevier, Amsterdam, 2003. [22] F. Hecht,New development in freefem++, J. Numer. Math., 20 (2012), pp. 251--265. [23] M. Hinterm\" uller, T. Surowiec, and A. K\" ammler,Generalized Nash equilibrium problems in Banach spaces: Theory, Nikaido-Isoda-based path-following methods and applications, SIAM J. Optim., 25 (2015), pp. 1826--156. [24] S. Labb\' e and E. Tr\' elat,Uniform controllability of semi-discrete approximations of parabolic control systems, Systems Control Lett., 55 (2006), pp. 597--609. [25] J.-L. Lions,Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), pp. 1--68. [26] S. Moya and J. Escobar,Stackelberg-Nash equilibrium in traffic control problem on an intersection with a priority road, IMA J. Math. Control Inform. 32 (2015), pp. 161--194, doi:10.1093/imamci/dnt036. [27] A. M\" unch and D.A. Souza,A mixed formulation for the direct approximation of l2- weighted controls for the linear heat equation, Adv. Comput. Math., 42 (2016), http: //hal.archives-ouvertes.fr/hal-00998765. [28] J.E. Roberts and J.-M. Thomas,Mixed and hybrid methods, in Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991, pp. 523--639. [29] D.L. Russell,Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), pp. 639--739. Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. A2700 PIT\' AGORAS P. DE CARVALHO AND ENRIQUE FERN\' ANDEZ-CARA [30] S.P. Sethi,Optimal Control Theory. Applications to Management Science and Economics, 3rd ed., Springer, Cham, 2019. [31] V. Ungureanu,Pareto-Nash-Stackelberg Game and Control Theory. Intelligent Paradigms and Applications, Smart Innovation Systems Technologies 89, Springer, Cham, 2018. [32] E. Zuazua,Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev., 47 (2005), pp. 197--243. [33] E. Zuazua,Controllability and observability of partial differential equations: Some results and open problems, in Handbook of Differential Equations: Evolutionary Equations, Vol. III, Elsevier/North-Holland, Amsterdam, 2007, pp. 527--621. Downloaded 11/11/22 to 52.18.63.169 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy