Abstract
A class of infinite horizon optimal control problems subject to semilinear parabolic equations is investigated. First and second order optimality conditions are obtained, in the presence of constraints on the controls, which can be either pointwise in space-time, or pointwise in time and \(L^2\) in space. These results rely on a new \(L^\infty \) estimate for nonlinear parabolic equations in an essential manner.
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1 Introduction
We study the optimal control problem
where \(\nu > 0\), \(y_d \in L^2(\varOmega \times (0,\infty )) \cap L^p(0,\infty ;L^2(\varOmega ))\) with \(p \in (\frac{4}{4 - n},\infty ]\), and
Above \({K_{ad}}\) denotes a closed, convex, and bounded set in \(L^2(\omega )\), and \(y_u\) is the solution of the following parabolic equation:
Here \(\varOmega \) is a bounded domain in \({\mathbb {R}}^n\), \(1 \le n \le 3\), with a Lipschitz boundary \(\varGamma \), and \(\varOmega \) is an interval if \(n = 1\), \(\omega \) is a measurable subset of \(\varOmega \) with positive Lebesgue measure, \(\chi _\omega \) denotes the characteristic function of \(\omega \), \(a \in L^\infty (\varOmega )\), \(0 \le a \not \equiv 0\), \(g \in L^2(Q)\), and additionally \(g \in L^p(0,\infty ;L^2(\varOmega ))\) with \(p \in (\frac{4}{4 - n},\infty ]\) if \(n = 2\) or 3, and \(y_0 \in L^\infty (\varOmega )\). For every \(u \in {U_{ad}}\), the symbol \(u\chi _\omega \) is defined as follows:
Possible choices for \({K_{ad}}\) include
Concerning the nonlinearity \(f:Q\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) we assume that it is a Carathéodory function of class \(C^1\) with respect to the last variable satisfying the following properties:
for almost all \((x,t) \in Q\). Let us observe that (1.5) and (1.6) imply
Moreover, (1.4) and (1.6) along with the mean value theorem yield
The following generalized Poincaré inequality will frequently be used
All along this paper we will assume that
Remark 1.1
The operator \(-\varDelta \) can be replaced by any uniformly elliptic operator with \(L^\infty (\varOmega )\) coefficients. The assumption (1.4) can be relaxed by assuming that \(f(\cdot ,\cdot ,0) \in L^2(Q) \cap L^\infty (0,\infty ;L^2(\varOmega ))\) and then redefining f and g as \(f(x,t,y) - f(x,t,0)\) and \(g(x,t) - f(x,t,0)\), respectively.
By investigating (P) we continue our efforts on studying infinite horizon optimal control problems with semilinear parabolic equations as constraints. In [8] the nonlinearities were chosen of polynomial type, no constraints were enforced on the controls, and the focus was put on nonsmooth, sparsity enhancing control costs, which entail that the controls settle down at zero once the states enter into a neighborhood of a stable equilibrium. Later, in [9] the nonlinearity was not restricted to be a polynomial and the conditions on f were very similar to those imposed in the present paper. The same type of control constraints were imposed as well. The major step forward in the current paper compared to [9] consists in an \(L^\infty (Q)\) estimate of the states for feasible controls, i.e. for controls with the property that the associated states \(y_u\) are in \(L^2(Q)\). Utilizing this property, well-posedness and \(C^2\) regularity of the control-to-state mapping, associating the infinite horizon controls to the infinite horizon states, can be guaranteed, and a second order analysis of (P) becomes possible. This was not the case in [9], where the first order conditions of the infinite horizon problem were obtained as the limit of the associated finite horizon problems, and no second order analysis was carried out. The authors are not aware about the availability of the second order analysis for optimal control problems with constraints as in (1.2) even in the finite horizon case. Along a related, but different line of research we also investigated infinite horizon optimal control problems with a discount factor on the state, [11] and [12]. This allows to treat a larger class of nonlinearities at the expense of less information of the optimal states as time increases.
Most of the literature on infinite horizon problems is carried out for ordinary differential equations. Let us mention some of these contributions. In [7] the importance of infinite horizon problems in applications is stressed. In general, when formulating optimal control problems, the time horizon can be subject to ambiguity. In such cases the choice as infinite horizon problem can be a valuable choice. The first article, focusing on infinite horizon problems may be [15]. More recent contributions all in the context of ordinary differential equations are available for instance in [1, 2, 4]. Concerning the literature, pointwise constraints as in (1.3) have received considerably more attention than norm constraints as in (1.2). However, from a practical point of view (1.2) appears to be equally important. In the case of optimal control of Navier–Stokes equations the suitability of this type of constraints was discussed in [14]. The use of the \(L^1(\omega )\) norm replacing the \(L^2(\omega )\) was studied in [10]. The last two references were devoted to final horizon control problems.
Briefly, the paper is structured in the following way. In Sect. 2, the existence of optimal controls and first order optimality conditions are established. Necessary and sufficient second order conditions for the two choices of \({K_{ad}}\) in (1.2) and (1.3) are obtained in Sect. 3. Section 4 is devoted to convergence results for the finite horizon problems associated to (P), to the infinite horizon problem. This is not only of intrinsic interest but also of relevance for numerical realization. In the Appendix the relevant results for the state equation, and the associated linearized and adjoint equations are established. The \(L^\infty (Q)\) regularity result for the state equation, already mentioned above, may be of interest beyond its application in optimal control.
2 Existence of an optimal control and first order optimality conditions
In this section, we prove the existence of an optimal solution of (P) and derive the first order optimality conditions satisfied by any local minimizer. For this purpose we will also address the issue of differentiability of the relation control-to-state and of the cost functional J. Let us observe that Theorem A.2 implies the existence of a unique state \(y_u\) for every control \(u \in {U_{ad}}\). However, it could happen that \(y_u \not \in L^2(Q)\) and, consequently, \(J(u) = \infty \). Therefore, the assumption about the existence of a control \(u_0 \in {U_{ad}}\) such that \(J(u_0) < \infty \) is needed. This issue will not be addressed in this paper, the reader is referred, for instance, to [3] and [8] for this question. We will say that u is a feasible control if \(u \in {U_{ad}}\) and \(J(u) < \infty \).
For \(0 < T \le \infty \) we set \(W(0,T) = \{y \in L^2(0,T;H^1(\varOmega )): \frac{\partial y}{\partial t} \in L^2(0,T;H^1(\varOmega )^*)\} \) with \(\Vert y\Vert _{W(0,T)} = \Big (\Vert y\Vert _{L^2(0,T;H^1(\varOmega ))}^2 + \Big \Vert \frac{\partial y}{\partial t}\Big \Vert _{L^2(0,T;H^1(\varOmega )^*)}^2\Big )^{\frac{1}{2}}\) as norm. It is well known that \((W(0,T),\Vert \cdot \Vert _{W(0,T)})\) is a Banach space. In fact, it is a Hilbert space because \(\Vert \cdot \Vert _{W(0,T)}\) is a Hilbertian norm. Furthermore, the embedding \(W(0,T) \subset C([0,T];L^2(\varOmega ))\) is continuous for \(T \le \infty \) and W(0, T) is compactly embedded in \(L^2(0,T;L^2(\varOmega ))\) if \(T < \infty \).
Theorem 2.1
Let us assume that there exists a feasible control \(u_0\). Then, (P) has at least one solution.
Proof
Let \(\{u_k\}_{k = 1}^\infty \subset {U_{ad}}\) be a minimizing sequence of feasible controls with associated states \(\{y_{u_k}\}_{k = 1}^\infty \). Since \(J(u_k) \rightarrow \inf \hbox {(P)} < \infty \), then the boundedness of \(\{u_k\}_{k = 1}^\infty \) and \(\{y_{u_k}\}_{k = 1}^\infty \) in \(L^2(Q_\omega )\) and \(L^2(Q)\), respectively, follows. Then, taking subsequences we can assume that \((u_k,y_{u_k}) \rightharpoonup (\bar{u},\bar{y})\) in \(L^2(Q_\omega ) \times L^2(Q)\). Since \({U_{ad}}\) is a closed and convex subset of \(L^2(Q_\omega )\), we infer that \(\bar{u} \in {U_{ad}}\). Due to the weak lower semicontinuity of J with respect to (y, u) in \(L^2(Q) \times L^2(Q_\omega )\), it is enough to establish that \(\bar{y}\) is the state associated to \(\bar{u}\) to conclude the proof. For this purpose we have to show that \(\bar{y}\) satisfies (A.2) with \(g + \chi _\omega \bar{u}\) on the right hand side for every \(T < \infty \). The only delicate point in this respect is to prove the convergence of \(f(x,t,y_{u_k}) \rightarrow f(x,t,\bar{y})\) in \(L^2(Q_T)\) for every \(T > 0\), where \(Q_T = \varOmega \times (0,T)\). Using the boundedness of \(\{(u_k,y_{u_k})\}_{k = 1}^\infty \) in \([L^2(Q_\omega )\cap L^\infty (0,\infty ;L^2(\omega ))] \times L^2(Q)\) we deduce from (A.4)–(A.6) the boundedness of \(\{y_{u_k}\}_{k = 1}^\infty \) in \(W(0,\infty ) \cap L^\infty (Q)\) and \(\{f(\cdot ,\cdot ,y_{u_k})\}_{k = 1}^\infty \) in \(L^\infty (Q) \cap L^2(Q)\). Hence, using the compactness of the embedding \(W(0,T) \subset L^2(Q_T)\) the desired convergence follows. \(\square \)
Hereafter, the following additional hypothesis on f is assumed:
Let us denote for every p satisfying (1.10)
and by \(G_p:\mathcal {U}_p \longrightarrow Y\) the mapping \(G_p(u) = y_u\), where \(y_u\) is the solution of (1.1). \(Y_p\) is a Banach space when endowed with the associated graph norm. We observe that \(\mathcal {U}_\infty \subset \mathcal {U}_p\) and \(G_\infty \) is the restriction of \(G_p\) to \(\mathcal {U}_\infty \).
Theorem 2.2
Let us assume that \(\mathcal {U}_p\) is not empty. Then, \(\mathcal {U}_p\) is an open subset of \(L^2(Q_\omega ) \cap L^p(0,\infty ;L^2(\omega ))\) and the mapping \(G_p\) is of class \(C^1\). Moreover, given \(u \in \mathcal {U}_p\) and \(v \in L^2(Q_\omega ) \cap L^p(0,\infty ;L^2(\omega ))\), \(z_v = DG_p(u)v\) is the unique solution of
Proof
The proof will be based on the implicit function theorem. For this purpose we define the mapping
By definition of \(Y_p\) and using (1.8), we deduce that \(\mathcal {F}_p\) is well defined and is of class \(C^1\). Further, we have that \(\mathcal {F}_p(y_u,u) = (0,0)\) for every \(u \in \mathcal {U}_p\) and
Then, \(\frac{\partial \mathcal {F}_p}{\partial y}(y_u,u)\) is an isomorphism if and only if the equation
has a unique solution in \(Y_p\) for every \((h,z_0) \in L^2(Q) \cap L^p(0,\infty ;L^2(\varOmega )) \times L^\infty (\varOmega )\) with continuous dependence. This is an immediate consequence of Theorem A.3 with \(d(x,t,s) = \frac{\partial f}{\partial y}(x,t,s)\) and \(y = y_u \in L^\infty (Q)\). Finally, the theorem follows by applying the implicit function theorem. \(\square \)
As a consequence of the above theorem, we have that \(J:\mathcal {U}_p \longrightarrow {\mathbb {R}}\) is well defined. The next theorem establishes its differentiability.
Theorem 2.3
Assuming that \(\mathcal {U}_p\) is not empty, the functional J is of class \(C^1\) and for every \(u \in \mathcal {U}_p\) and \(v \in L^2(Q_\omega ) \cap L^p(0,\infty ;L^2(\omega ))\) its derivative is given by
where \(z_{u,v} = G_p'(u)v\) and \(\varphi _u \in W(0,\infty ) \cap L^\infty (Q)\) satisfies
The fact that J is of class \(C^1\) is an immediate consequence of Theorem 2.2 and the chain rule. Formula (2.4) is deduced in the standard way from Eqs. (2.2) and (2.5). Concerning the well posedness of (2.5) we refer to Theorem A.4.
We conclude this section establishing the first order optimality conditions satisfied by every local minimizer of (P) and deducing some consequences from them. In this paper, a local minimizer \(\bar{u}\) is understood in the \(L^2(Q_\omega )\) sense and it is assumed that \(\bar{u} \in \mathcal {U}_\infty \cap {U_{ad}}\).
Theorem 2.4
Let \(\bar{u}\) be a local minimizer of (P). Then, there exist \(\bar{y}, \bar{\varphi }\in W(0,\infty ) \cap L^\infty (Q)\) such that
This theorem is an immediate consequence of Theorem 2.3 and the inequality \(J'(\bar{u})(u - \bar{u}) \ge 0\) for all \(u \in {U_{ad}}\).
Corollary 2.1
Let \(\bar{\varphi }\) and \(\bar{u}\) satisfy (2.7) and (2.8). If \({K_{ad}}\) is given by (1.2), then the following properties hold for almost all \(t \in (0,\infty )\)
In the case that \({K_{ad}}\) is given by (1.3), then we have
In both cases we have that \(\bar{u} \in L^\infty (Q_\omega )\).
Proof
For the proof of (2.9) and (2.10) the reader is referred to [9, Lemma 3.2]. Let us prove (2.11). First, we assume that \(\Vert \bar{\varphi }(t) + \nu \bar{u}(t)\Vert _{L^2(\omega )} \ne 0\). Then, again from [9, Lemma 3.2] we obtain
This yields
Taking the norm in \(L^2(\omega )\) in the above expression and using that \(\Vert \bar{u}(t)\Vert _{L^2(\omega )} = \gamma \) we infer
Identities (2.14) and (2.15) imply (2.11). In the case \(\Vert \bar{\varphi }(t) + \nu \bar{u}(t)\Vert _{L^2(\omega )} = 0\) and \(\Vert \bar{u}(t)\Vert _{L^2(\omega )} = \gamma \) we have that
Therefore, (2.11) also holds. Let us prove (2.12). If \(\Vert \bar{u}(t)\Vert _{L^2(\omega )} < \gamma \), then (2.10) implies that \(|\bar{u}(x,t)| = \frac{1}{\nu }|\bar{\varphi }(x,t)| \le \frac{1}{\nu }\Vert \bar{\varphi }\Vert _{L^\infty (Q_\omega )}\). If \(\Vert \bar{u}(t)\Vert _{L^2(\omega )} = \gamma \), the inequality \(\Vert \bar{\varphi }(t)\Vert _{L^2(\omega )} \ge \gamma \nu \) follows from (2.15) and (2.16). Then, (2.11) implies that \(|\bar{u}(x,t)| \le \frac{1}{\nu }\Vert \bar{\varphi }\Vert _{L^\infty (Q_\omega )}\).
Finally, the identity (2.13) is well known.
\(\square \)
3 Second order optimality conditions
In this section we address the second order optimality conditions for (P). For this purpose, in addition to assumptions (1.4)–(1.7) we impose the following hypotheses: \(f:Q \times {\mathbb {R}} \longrightarrow {\mathbb {R}}\) is of class \(C^2\) with respect to the second variable and satisfies
for almost all \((x,t) \in Q\). We observe that (3.1) and (3.2) imply (2.1). Indeed, it is enough to select
Then, using the mean value theorem we infer for almost all \((x,t) \in Q\)
Theorem 3.1
Under assumptions (1.4)–(1.7) and (3.1)–(3.2) and supposing that \(\mathcal {U}_p\) is not empty, \(G_p:{\mathcal {U}_p} \longrightarrow Y_p\) is of class \(C^2\). Moreover, given \(u \in \mathcal {U}_p\) and \(v_1, v_2 \in L^2(Q_\omega ) \cap L^p(0,\infty ;L^2(\omega ))\), then \(z_{v_1,v_2} = G_p''(u)(v_1,v_2)\) is the solution of the equation
where \(z_{v_i} = G_p'(u)v_i\) for \(i = 1, 2\).
The \(C^2\) differentiability of G follows from the implicit function theorem applied to the mapping \(\mathcal {F}_p\) introduced in the proof of Theorem 2.2. It is enough to observe that now \(\mathcal {F}_p\) is of class \(C^2\). The Eq. (3.4) follows differentiating the identity \(\mathcal {F}_p(G_p(u),u) = 0\) twice.
As a consequence of Theorem 3.1 and the chain rule we have the following corollary.
Corollary 3.1
If \(\mathcal {U}_p\) is not empty, then the function \(J:\mathcal {U}_p \longrightarrow {\mathbb {R}}\) is of class \(C^2\) and we have
for every \(u \in \mathcal {U}\) and \(v_1, v_2 \in L^2(Q_\omega ) \cap L^p(0,\infty ;L^2(\omega ))\).
Remark 3.1
Under assumptions (1.4)–(1.7) and (3.1)–(3.2), for every \(u \in \mathcal {U}_p\) the linear form \(J'(u):L^2(Q_\omega ) \cap L^p(0,\infty ;L^2(\omega )) \longrightarrow {\mathbb {R}}\) as well as the bilinear form \(J''(u):[L^2(Q_\omega ) \cap L^p(0,\infty ;L^2(\omega ))]^2 \longrightarrow {\mathbb {R}}\) can be extended to continuous linear and bilinear forms \(J'(u):L^2(Q_\omega ) \longrightarrow {\mathbb {R}}\) and \(J''(u):L^2(Q_\omega )^2 \longrightarrow {\mathbb {R}}\) given by the same expressions (2.4) and (3.5), respectively. Indeed, this is an immediate consequence of Theorem A.3 along with the \(L^\infty (Q) \cap L^2(Q)\) regularity of the adjoint states established in Theorem A.4.
The analysis of second order optimality conditions is carried out in the next two subsections, where we consider the cases with \({K_{ad}}\) given by (1.2) or (1.3).
3.1 Case I: \({K_{ad}}= B_\gamma = \{v \in L^2(\omega ): \Vert v\Vert _{L^2(\omega )} \le \gamma \}\).
For this case we consider the Lagrange function
Theorem 2.3 and Corollary 3.1 imply that \(\mathcal {L}\) is of class \(C^2\) and we have the expressions
The identities (3.6) and (3.7) define continuous linear and bilinear forms on \(L^2(Q_\omega )\) and \(L^2(Q_\omega )^2\), respectively.
Let \(\bar{u} \in {U_{ad}}\cap \mathcal {U}_\infty \) satisfy the first oder optimality conditions (2.6)–(2.8). Associated with \(\bar{u}\) we define \(\bar{\lambda }(t) = \Vert \bar{\varphi }(t) + \nu \bar{u}(t)\Vert _{L^2(\omega )}\). From Theorem 2.4 and (2.12) we get that \(\bar{\lambda }\in L^\infty (0,\infty ) \cap L^2(0,\infty )\). We also set
The choice of \(\bar{\lambda }\) as Lagrange multiplier associated with the control constraint is suggested by (2.10). Actually, next lemma confirms that this is the correct choice.
Lemma 3.1
Let \(\bar{u}\) and \(\bar{\varphi }\) satisfy (2.7) and (2.8). Then we have \(\frac{\partial \mathcal {L}}{\partial u}(\bar{u},\bar{\lambda })v = 0\) for every \(v \in L^2(Q_\omega )\).
Proof
Using (3.6), (2.10), (2.11), and (2.15) we infer
\(\square \)
In order to formulate the second order optimality conditions we introduce the cone of critical directions associated with \(\bar{u}\):
Then we have the following second order necessary optimality conditions.
Theorem 3.2
If \(\bar{u}\) is a local minimizer of (P), then \(\frac{\partial \mathcal {L}}{\partial u}(\bar{u},\bar{\lambda })v^2 \ge 0\) for all \(v \in C_{\bar{u}}\).
Proof
Since \(\bar{u}\) is a local minimizer of (P), there exists \(\varepsilon > 0\) such that \(J(\bar{u}) \le J(u)\) for all \(u \in {U_{ad}}\cap B_\varepsilon (\bar{u})\), where \(B_\varepsilon (\bar{u}) = \{u \in L^2(Q_\omega ): \Vert u - \bar{u}\Vert _{L^2(Q_\omega )} < \varepsilon \}\). Due to \(\bar{u} \in \mathcal {U}_\infty \) and since \(\mathcal {U}_\infty \) is an open subset of \(L^2(Q_\omega ) \cap L^\infty (0,\infty ;L^2(\omega ))\), we can select \(\varepsilon \) small enough so that every control \(u \in B_\varepsilon (\bar{u})\) satisfying \(\Vert u - \bar{u}\Vert _{L^\infty (0,\infty ;L^2(\omega ))} < \varepsilon \) belongs to \(\mathcal {U}_\infty \).
Let \(v \in C_{\bar{u}} \cap L^\infty (0,\infty ;L^2(\omega ))\). The assumption \(v \in L^\infty (0,\infty ;L^2(\omega ))\) will be removed later. Let us fix an integer
and set
It is obvious that \(\{v_k\}_{k \ge k_0} \subset L^2(Q_\omega ) \cap L^\infty (0,\infty ;L^2(\omega ))\). Moreover, the convergence \(v_k \rightarrow v\) in \(L^2(Q_\omega )\) follows from Lebesgue’s dominated convergence theorem.
For fixed \(k \ge k_0\), we define
and \(\phi _k:(-\alpha _k,+\alpha _k) \longrightarrow L^2(Q_\omega ) \cap L^\infty (0,\infty ;L^2(\omega ))\) by
By definition of \(\alpha _k\) we have \(\frac{\rho ^2}{\gamma ^2}\Vert v_k(t)\Vert ^2_{L^2(\omega )} < 1\) for all \(k \ge k_0\), \(|\rho | < \alpha _k\), and almost all \(t \in (0,\infty )\). Moreover, \(|\phi _k(\rho )| \le |\bar{u}| + \frac{\varepsilon }{2\Vert v\Vert _{L^2(Q_\omega )}}|v| \in L^2(Q_\omega ) \cap L^\infty (0,\infty ;L^2(\omega ))\). Hence, the mapping \(\phi _k\) is well defined and it is of class \(C^\infty \). Let us prove some properties of this mapping.
I - \(\phi _k(\rho ) \in {U_{ad}}\) for all \(\rho \in [0,+\alpha _k)\). Let us set \(u_\rho = \phi _k(\rho )\). Then, we have for almost all \(t \in (0,\infty )\)
In the case \(t \in I_\gamma \), we have \(v_k(t) = v(t)\). Then, using that \(v \in C_{\bar{u}}\) we deduce that the last integral in the above inequality is less than or equal to zero and, consequently, (3.8) leads to \(\Vert u_\rho (t)\Vert ^2_{L^2(\omega )} \le \gamma ^2\). If \(\gamma ^2 - \frac{1}{k}< \Vert \bar{u}(t)\Vert ^2_{L^2(\omega )} < \gamma ^2\), then we have \(v_k(t) = 0\) by definition and, hence, (3.8) implies that \(\Vert u_\rho (t)\Vert ^2_{L^2(\omega )} \le \gamma ^2\). Finally, we assume that \(\Vert \bar{u}(t)\Vert ^2_{L^2(\omega )} \le \gamma ^2 - \frac{1}{k}\). Then, we infer from the definition of \(\alpha _k\)
II - \(\Vert \phi _k(\rho ) - \bar{u}\Vert _{L^2(Q_\omega )} \le \varepsilon \). From the definition of \(\phi _k\) we get
From the definition of \(\alpha _k\) and \(k \ge k_0 > \frac{1}{\gamma ^2}\) we obtain
Moreover, \(\alpha _k \le \frac{\varepsilon }{2\Vert v\Vert _{L^2(Q_\omega )}}\) holds. Then, we have
The last inequality is consequence of \(k \ge k_0 > \sqrt{\frac{2\Vert \bar{u}\Vert _{L^2(Q_\omega )}}{\gamma ^4\varepsilon }}\).
III - \(\phi _k(\rho ) \in \mathcal {U}_\infty \). Arguing as in the previous step and using again the definition of \(\alpha _k\) and \(k_0\) with \(\Vert \bar{u}\Vert _{L^2(Q_\omega )}\) and \(\Vert v\Vert _{L^2(Q_\omega )}\) replaced by \(\Vert \bar{u}\Vert _{L^\infty (0,\infty ;L^2(\omega ))}\) and \(\Vert v\Vert _{L^\infty (0,\infty ;L^2(\omega ))}\), respectively, we infer that \(\Vert \phi _k(\rho ) - \bar{u}\Vert _{L^\infty (0,\infty ;L^2(\omega ))} < \varepsilon \). Due to the choice of \(\varepsilon \) this implies that \(\phi _k(\rho ) \in \mathcal {U}_\infty \).
Now we define the function \(\psi _k:(-\alpha _k,+\alpha _k) \longrightarrow {\mathbb {R}}\) by \(\psi _k(\rho ) = J(\phi _k(\rho ))\). From the local optimality of \(\bar{u}\) and the established properties of \(\phi _k\) we infer that \(\psi _k(0) = J(\bar{u}) \le J(\phi _k(\rho )) = \psi _k(\rho )\) for every \(\rho \in [0,+\alpha _k)\). Since \(\psi _k\) is of class \(C^2\), and \(\psi '_k(0) = 0\) then \(\psi _k''(0) \ge 0\). Hence, we get
Using (2.10), (2.11), (2.15), and (2.16) we obtain that
Inserting this in the above inequality we infer with (3.7)
Now, the convergence \(v_k \rightarrow v\) in \(L^2(Q)\) implies
Finally, we remove the assumption \(v \in L^\infty (0,\infty ;L^2(\omega ))\). Given \(v \in C_{\bar{u}}\), we define \(v_k(x,t) = \frac{v(x,t)}{1 + \frac{1}{k}\Vert v(t)\Vert _{L^2(\omega )}}\) for every integer \(k \ge 1\). Then, we have \(\{v_k\}_{k = 1}^\infty \subset L^\infty (0,\infty ;L^2(\omega )) \cap L^2(Q_\omega )\) and \(v_k \rightarrow v\) in \(L^2(Q_\omega )\). Using that \(v \in C_{\bar{u}}\) we get
Identity (2.11) implies
Therefore from (2.10) and the above relations we deduce
Hence, \(\{v_k\}_{k = 1}^\infty \subset C_{\bar{u}} \cap L^\infty (0,\infty ;L^2(\omega ))\) holds and, consequently, \(\frac{\partial ^2L}{\partial u^2}(\bar{u},\bar{\lambda })v_k^2 \ge 0\) for all \(k \ge 1\). Finally, passing to the limit as \(k \rightarrow \infty \) we conclude that \(\frac{\partial ^2\,L}{\partial u^2}(\bar{u},\bar{\lambda })v^2 \ge 0\). \(\square \)
Now, we give a second order sufficient optimality condition.
Theorem 3.3
Let \(\bar{u} \in {U_{ad}}\cap \mathcal {U}_\infty \) satisfy the first order optimality conditions (2.6)–(2.8) and the second order condition \(\frac{\partial ^2\mathcal {L}}{\partial u^2}(\bar{u},\mu )v^2 > 0\) for every \(v \in C_{\bar{u}} {\setminus } \{0\}\). Then, there exists \(\kappa > 0\) and \(\varepsilon > 0\) such that
Proof
We argue by contradiction and assume that (3.9) does not hold. Then, for every integer \(k \ge 1\) there exists a control \(u_k \in {U_{ad}}\) such that
We define \(v_k = \frac{1}{\rho _k}(u_k - \bar{u})\). Since \(\Vert v_k\Vert _{L^2(Q_\omega )} = 1\) for every k, taking a subsequence, we can assume that \(v_k \rightharpoonup v\) in \(L^2(Q_\omega )\). From (3.10) we deduce that \(\{y_{u_k}\}_{k = 1}^\infty \) is a bounded sequence in \(L^2(Q)\), hence \(\{u_k\}_{k=1}^\infty \subset \mathcal {U}_\infty \). Moreover, given \(p \in (\frac{4}{4 - n},\infty )\) we have
Then, \(y_{u_k} = G_p(u_k) \rightarrow G_p(\bar{u}) = \bar{y}\) in \(Y_p\). Consequently, there exists a ball \(B_r(\bar{u}) \subset L^2(Q_\omega ) \cap L^p(0,\infty ;L^2(\omega ))\) and \(k_0 \ge 1\) such that \(\{u_k\}_{k \ge k_0} \subset \mathcal {U}_p\). The rest of the proof is split into three steps.
Step I \(v \in C_{\bar{u}}\). From (2.4) and (2.8) we infer that
Using the differentiability of the mapping \(J:\mathcal {U}_p \longrightarrow {\mathbb {R}}\) we infer with the mean value theorem and (3.10)
where \(\theta _k \in [0,1]\), \(u_{\theta _k} = \bar{u} + \theta _k(u_k - \bar{u})\), and \(\varphi _{\theta _k}\) is the adjoint state corresponding to \(u_{\theta _k}\). Since \(y_{\theta _k}= G_p(u_{\theta _k}) \rightarrow G_p(\bar{u}) = \bar{y}\) in \(Y_p\), we deduce from Theorem A.4 that \(\varphi _{\theta _k} \rightarrow \bar{\varphi }\) in \(Y_p\) as \(k \rightarrow \infty \). Then, it is straightforward to pass to the limit in the above expression and to get \(J'(\bar{u})v \le 0\). This inequality and (3.11) imply that \(J'(\bar{u})v = 0\).
Next, taking into account that \(\Vert u_k(t)\Vert _{L^2(\omega )} \le \gamma \) for almost all \(t > 0\), we have for almost every \(t \in I_\gamma \)
We define the function \(\phi \in L^\infty (0,\infty )\) by \(\phi (t) = 1\) if \(\int _\omega \bar{u}(t)v(t)\,\textrm{d}x> 0\) and 0 otherwise. Then, from the convergence \(v_k \rightharpoonup v\) in \(L^2(Q_\omega )\) and the fact that \(\phi \bar{u} \in L^2(Q_\omega )\) we infer from the above inequality
This is possible if and only if \(\int _\omega \bar{u}(t)v(t)\,\textrm{d}x\le 0\) for almost all \(t \in I_\gamma \). Finally, we prove that this integral is 0 if \(t \in I_\gamma ^+\). For this purpose we use Lemma 3.1, (3.6), and the fact that \(J'(\bar{u})v = 0\) as follows
which implies that \(\int _\omega \bar{u}(t)v(t)\,\textrm{d}x= 0\) for almost all \(t \in I^+_\gamma \), and thus \(v \in C_{\bar{u}}\).
Step II \(\frac{\partial ^2\mathcal {L}}{\partial u^2}(\bar{u},\bar{\lambda })v^2 \le 0\). First we observe that
This inequality and (3.10) imply
Performing a Taylor expansion and using again Lemma 3.1 we infer for some \(\vartheta _k \in [0,1]\)
Dividing the above inequality by \(\frac{\rho _k^2}{2}\) we get
Denoting by \(u_{\vartheta _k} = \bar{u} + \vartheta _k(u_k - \bar{u})\), \(y_{\vartheta _k}\) its associated state, and \(\varphi _{\vartheta _k}\) the corresponding adjoint state, we get from (3.7)
where \(z_{\vartheta _k,v_k}\) satisfies the equation
Now, we study the lower limit of (3.12). From Theorem A.3 and the boundedness of \(\{v_k\}_{k = 1}^\infty \) and \(\{y_{\vartheta _k}\}_{k = 1}^\infty \) in \(L^2(Q_\omega )\) and \(L^\infty (Q)\), respectively, we infer the boundedness of \(\{z_{\vartheta _k,v_k}\}_{k = 1}^\infty \) in \(W(0,\infty )\). Therefore, we can extract a subsequence, that we denote in the same way, such that \(\{z_{\vartheta _k,v_k}\}_{k = 1}^\infty \) converges weakly in \(W(0,\infty )\). Moreover, the convergence \(u_{\vartheta _k} \rightarrow \bar{u}\) in \(L^p(0,\infty ;L^2(\omega ))\) implies \(y_{\vartheta _k} = G_p(u_{\vartheta _k}) \rightarrow G_p(\bar{u}) = \bar{y}\) in \(Y_p\). Using this and the convergence \(v_k \rightharpoonup v\) in \(L^2(Q_\omega )\), it is straightforward to pass to the limit in (3.14) and to deduce that \(z_{\vartheta _k,v_k} \rightharpoonup z_v\) in \(W(0,\infty )\), where \(z_v\) is the solution of (2.2). Further, the convergence of \(y_{\vartheta _k} \rightarrow \bar{y}\) in \(Y_p\) implies the convergence in \(L^p(0,\infty ;L^2(\varOmega )) \cap L^\infty (Q)\). Then, from Theorem A.4 we infer that \(\varphi _{\vartheta _k} \rightarrow \bar{\varphi }\) in \(W(0,\infty ) \cap L^\infty (Q)\). Indeed, subtracting the equations satisfied by \(\varphi _{\vartheta _k}\) and \(\bar{\varphi }\) we get for \(\psi _k = \varphi _{\vartheta _k} - \bar{\varphi }\)
Then, using (3.3), the established convergence \(y_{\vartheta _k} \rightarrow \bar{y}\), (A.21), and (A.22) we get the claimed convergence of \(\{\varphi _{\vartheta _k}\}_{k = 1}^\infty \) to \(\bar{\varphi }\).
Now, we take the lower limit in (3.12). For this purpose we take into account that \(z_{\vartheta _k,v_k} \rightharpoonup z_v\) in \(L^2(Q)\), \(v_k \rightharpoonup v\) in \(L^2(Q_\omega )\), and \(\bar{\lambda }\in L^\infty (Q)\) with \(\bar{\lambda }(t) \ge 0\) for almost all \(t \in (0,\infty )\). Hence, we get by (3.12)
Below we prove that
Thus, (3.7) and (3.15)–(3.16) yield \(\frac{\partial ^2\mathcal {L}}{\partial u^2}(\bar{u},\lambda )v^2 \le 0\).
Let us prove (3.16). Given \(\varepsilon > 0\), (2.7) implies the existence of \(T_\varepsilon > 0\) such that \(\Vert \bar{\varphi }(t)\Vert _{L^2(\varOmega )} < \varepsilon \) for every \(t \ge T_\varepsilon \). Further, the convergence \(z_{\vartheta _k,v_k} \rightharpoonup z_v\) in \(W(0,\infty )\) implies the convergence \(z_{\vartheta _k,v_k} \rightarrow z_v\) in \(L^2(Q_{T_\varepsilon })\). Using these properties and (3.2) with \(M = \Vert \bar{y}\Vert _{L^\infty (Q)}\) we get
The convergence \((y_{\vartheta _k},\varphi _{\vartheta _k}) \rightarrow (\bar{y},\bar{\varphi })\) in \(L^\infty (Q)^2\) and the boundedness of \(\{z_{\vartheta _k,v_k}\}_{k = 1}^\infty \) in \(W(0,\infty )\) imply that \(I_1 \rightarrow 0\) as \(k \rightarrow \infty \). The convergence \(z_{\vartheta _k,v_k} \rightarrow z_v\) in \(L^2(Q_{T_\varepsilon })\) implies that \(I_2 \rightarrow 0\) as well. For \(I_3\) we have
where we have used again the boundedness of \(\{z_{\vartheta _k,v_k}\}_{k = 1}^\infty \) in \(W(0,\infty )\). Since \(\varepsilon > 0\) is arbitrarily small, we deduce the convergence \(I_3 \rightarrow 0\) as \(k \rightarrow \infty \).
Step III—Final contradiction The facts proved in Steps I and II along with the assumption \(\frac{\partial ^2\mathcal {L}}{\partial u^2}(\bar{u},\bar{\lambda })v^2 > 0\) for every \(v \in C_{\bar{u}} {\setminus } \{0\}\) lead to \(v = 0\) and \(z_v = 0\). Therefore, looking at the relations (3.15) we obtain with (3.16) and \(\Vert v_k\Vert _{L^2(Q_\omega )} = 1\)
which contradicts the assumption \(\nu > 0\). \(\square \)
3.2 Case II: \({K_{ad}}= \{v \in L^2(\omega ): \alpha \le v(x) \le \beta \text { for a.a. } x \in \omega \}\).
In this case, the cone of critical directions is defined by
Analogously to Theorem 3.2 we have the following result.
Theorem 3.4
If \(\bar{u}\) is a local minimizer of (P), then \(J''(\bar{u})v^2 \ge 0\) for all \(v \in C_{\bar{u}}\).
Proof
Since \(\bar{u}\) is a local minimizer of (P), there exists \(\varepsilon > 0\) such that \(J(\bar{u}) \le J(u)\) for all \(u \in {U_{ad}}\cap B_\varepsilon (\bar{u})\), where \(B_\varepsilon (\bar{u}) = \{u \in L^2(Q_\omega ): \Vert u - \bar{u}\Vert _{L^2(Q_\omega )} < \varepsilon \}\). Given \(p \in \big (\frac{4}{4 - n},\infty )\) we have for every \(u \in {U_{ad}}\cap B_\varepsilon (\bar{u})\)
Therefore, we select \(\varepsilon > 0\) small enough, such that \({U_{ad}}\cap B_\varepsilon (\bar{u}) \subset \mathcal {U}_p\) holds. Now, given \(v \in C_{\bar{u}}\) we define for every integer \(k \ge 1\) the function \(v_k\) by
It is obvious that \(\{v_k\}_{k = 1}^\infty \subset L^\infty (Q_\omega ) \cap L^2(Q_\omega )\) and \(v_k \rightarrow v\) in \(L^2(Q_\omega )\) as \(k \rightarrow \infty \). Further, if we set \(\rho _k = \min \{\frac{1}{k^2},\frac{\beta - \alpha }{k},\frac{\varepsilon }{\Vert v\Vert _{L^2(Q_\omega )}}\}\), then \(\bar{u} + \rho v_k \in {U_{ad}}\cap B_\varepsilon (\bar{u})\) for every \(\rho \in (0,\rho _k)\). In view of (2.13), it is straightforward to check that the condition \(J'(\bar{u})v = 0\) in the definition of \(C_{\bar{u}}\) is equivalent to \((\bar{\varphi }+ \nu \bar{u})(x,t)v(x,t) = 0\) for almost all \((x,t) \in Q_\omega \). Using this fact, it is immediate that \(J'(\bar{u})v_k = 0\) for every k. Then, performing a Taylor expansion we get for every \(\rho \in (0,\rho _k)\)
Dividing by \(\frac{\rho ^2}{2}\) we deduce \(J''(\bar{u} + \theta _{\rho ,k}\rho v_k)v_k^2 \ge 0\). Since \(\bar{u} + \theta _{\rho ,k}\rho v_k \rightarrow \bar{u}\) in \(L^p(0,\infty ;L^2(\omega ))\) as \(\rho \rightarrow 0\), we deduce \(J''(\bar{u})v_k^2 \ge 0\). Moreover, since \(v_k \rightarrow v\) in \(L^2(Q_\omega )\) we infer from Theorem A.3 that \(z_{v_k} \rightarrow z_v\) in \(L^2(Q_\omega )\). Hence, we can pass to the limit in the previous inequality and obtain \(J''(\bar{u})v^2 \ge 0\). \(\square \)
Now, we establish the sufficient second order conditions for local optimality.
Theorem 3.5
Let \(\bar{u} \in {U_{ad}}\cap \mathcal {U}_\infty \) satisfy the first order optimality conditions (2.6)–(2.7) and the second order condition \(J''(\bar{u})v^2 > 0\) for every \(v \in C_{\bar{u}} {\setminus } \{0\}\). Then, there exists \(\kappa > 0\) and \(\varepsilon > 0\) such that
The proof of this theorem follows by contradiction similarly to the proof of Theorem 3.3 with the obvious simplifications due to the constraints under consideration in this second case for \({U_{ad}}\). For the proof of these results for finite horizon control problems the reader is also referred to [5, 13]. The difficulties resulting from the infinite horizon can be overcome by following the arguments used in the proof of Theorem 3.3.
4 Approximation by finite horizon problems
In this section we consider the approximation of (P) by finite horizon optimal control problems and provide error estimates for these approximations. For every \(0< T < \infty \) we consider the control problem
where \(U_{T,ad}= \{u \in L^2(Q_{T,\omega }): u(t) \in {K_{ad}}\text { for a.a. } t \in (0,T)\}\),
with \(Q_T = \varOmega \times (0,T)\), \(Q_{T,\omega } = \omega \times (0,T)\), and \(y_{T,u}\) denotes the solution of the equation
For every control \(u \in L^2(Q_{T,\omega })\) with associated state \(y_{T,u}\) and adjoint state \(\varphi _{T,u}\) we define extensions to \(Q_\omega \) and Q, denoted by \(\hat{u}\), \(\hat{y}_{T,u}\), and \(\hat{\varphi }_{T,u}\), by setting \((\hat{u},\hat{\varphi }_{T,u})(x,t) = (0,0)\) if \(t > T\) and \(\hat{y}_{T,u}\) is the solution of (1.1) associated with the extension \(\hat{u}\). In this section, we assume that \(0 \in {K_{ad}}\). Hence, if \(u \in U_{T,ad}\), then \(\hat{u} \in {U_{ad}}\) holds. Given a local minimizer \(u_T\) of \(({\hbox {P}}_{T}) \), we denote by \(y_T\) and \(\varphi _T\) its associated state and adjoint state, respectively. Then, \((u_T,y_T,\varphi _T)\) satisfies the optimality conditions established in Theorem 2.4 with Q and \(Q_\omega \) replaced by \(Q_T\) and \(Q_{T,\omega }\). As a consequence, Corollary 2.1 is also satisfied by \((u_T,y_T,\varphi _T)\) with the same changes.
In case \({U_{ad}}\) is given by (1.2), we define \(\lambda _T(t) = \Vert \varphi _T(t) + \nu u_T(t)\Vert _{L^2(\omega )}\) for \(t \in (0,T)\) and the Lagrange function
for every \(p \in (\frac{4}{4 - n},\infty ]\). Arguing as in Lemma 3.1 we also have
The next two theorems establish the convergence of the approximating problems \(({\hbox {P}}_{T}) \)to (P) as \(T \rightarrow \infty \).
Theorem 4.1
For every \(T > 0\) the control problem \(({\hbox {P}}_{T}) \)has at least one solution \(u_T\). If (P) has a feasible control \(u_0\), then the extensions \(\{\hat{u}_T\}_{T > 0}\) of any family of solutions are bounded in \(L^2(Q_\omega )\). Every weak limit \(\bar{u}\) in \(L^2(Q_\omega )\) of a sequence \(\{\hat{u}_{T_k}\}_{k = 1}^\infty \) with \(T_k \rightarrow \infty \) as \(k \rightarrow \infty \) is a solution of (P). Moreover, strong convergence \(\hat{u}_{T_k} \rightarrow \bar{u}\) in \(L^p(0,\infty ;L^2(\omega ))\) holds for every \(p \in [2,\infty )\).
Proof
Since \(U_{T,ad}\) is not empty, the existence of solution for \(({\hbox {P}}_{T}) \)is a classical result. Actually, one can easily adapt the existence proof of solution for (P) to \(({\hbox {P}}_{T}) \). We denote by \({\tilde{y}}_T\) the extension of \(y_T\) by zero in \(\varOmega \times (T,\infty )\). We point out that \({\tilde{y}}_T \ne \hat{y}_T\). Let \(y^0\) be the solution of (1.1) corresponding to \(u_0\). By definition of feasible control we have that \(J(u_0) < \infty \). Using the optimality of \(u_T\) we obtain
This proves the boundedness of \(\{\hat{u}_T\}_{T > 0}\) and \(\{\tilde{y}_T\}_{T > 0}\) in \(L^2(Q_\omega )\) and \(L^2(Q)\), respectively. Let \(\{(\hat{u}_{T_k},{\tilde{y}}_{T_k})\}_{k = 1}^\infty \) be a sequence with \(T_k \rightarrow \infty \) as \(k \rightarrow \infty \) converging weakly to \((\bar{u},\bar{y})\) in \(L^2(Q_\omega ) \times L^2(Q)\). Since \(\{\hat{u}_{T_k}\}_{k = 1}^\infty \subset {U_{ad}}\) and \({U_{ad}}\) is closed in \(L^2(Q_\omega )\) and convex, we infer that \(\bar{u} \in {U_{ad}}\). Moreover, we can apply Theorem A.2 to the Eq. (4.1) and deduce the existence of a constant \(M_1\) independent of k such that for all \(k\ge 1\)
From this estimate and (A.6) we get the existence of a constant \(M_2\) such that
The two above estimates and (4.1) imply that
for a constant independent of k. Using the convergence of \(\tilde{y}_k \rightharpoonup \bar{y}\) in \(L^2(Q)\), the compactness of the embedding \(W(0,T) \subset L^2(Q_T)\) for every \(T < \infty \), and the above estimate, it is obvious to pass to the limit in the equation
for each \(T_k \ge T\), and to deduce that \(\bar{y}\) is the solution of (4.1) associated to \(\bar{u}\) for arbitrary \(0< T < \infty \). This proves that \(\bar{y}\) is the solution of (1.1) corresponding to \(\bar{u}\). Further, since \(\bar{y} \in L^2(Q)\), we deduce that \(\bar{u} \in \mathcal {U}_\infty \). Let us prove that \(\bar{u}\) is a solution of (P). For every feasible control u of (P) we have
This proves that \(\bar{u}\) is a solution of (P). Moreover, replacing u by \(\bar{u}\) in the above inequalities we infer
This convergence along with the weak convergence \((\hat{u}_{T_k},{\tilde{y}}_{T_k}) \rightharpoonup (\bar{u},\bar{y})\) in \(L^2(Q_\omega ) \times L^2(Q)\) implies the strong convergence. Finally, for any \(p \in (2,\infty )\) we have
\(\square \)
Theorem 4.2
Let \(\bar{u}\) be a strict local minimizer of (P). Then, there exist \(T_0 \in (0,\infty )\) and a family \(\{u_T\}_{T > T_0}\) of local minimizers to \(({\hbox {P}}_{T}) \)such that the convergence \(\hat{u}_T \rightarrow \bar{u}\) in \(L^p(0,\infty ;L^2(\omega ))\) holds as \(T \rightarrow \infty \) for every \(p \in [2,\infty )\).
Proof
Since \(\bar{u}\) is a strict local minimizer of (P), there exists \(\rho > 0\) such that \(J(\bar{u}) < J(u)\) for every \(u \in {U_{ad}}\cap B_\rho (\bar{u})\) with \(u \ne \bar{u}\), where \(B_\rho (\bar{u})\) is the closed ball in \(L^2(Q_\omega )\) centered at \(\bar{u}\) and radius \(\rho > 0\). We consider the control problems
where \(B_{T,\rho }(\bar{u}) = \{u \in L^2(Q_{T,\omega }): \Vert u - \bar{u}\Vert _{L^2(Q_{T,\omega })} \le \rho \}\). Obviously \(\bar{u}\) is the unique solution of \(({\hbox {P}}_\rho ) \). Existence of a solution \(u_T\) of \(({\hbox {P}}_{T,\rho }) \)is straightforward. Then, arguing as in the proof of Theorem 4.1 and using the uniqueness of the solution of \(({\hbox {P}}_\rho ) \), we deduce the convergence \(\hat{u}_T \rightarrow \bar{u}\) in \(L^2(Q_\omega )\) as \(T \rightarrow \infty \). This implies the existence of \(T_0 > 0\) such that \(\Vert u_T - \bar{u}\Vert _{L^2(Q_{T,\omega )}} \le \Vert \hat{u}_T - \bar{u}\Vert _{L^2(Q_\omega )} < \rho \) for all \(T > T_0\). Hence, \(u_T\) is also a local minimizer of \(({\hbox {P}}_{T}) \)for \(T > T_0\). The strong convergence \(\hat{u}_T \rightarrow \bar{u}\) in \(L^p(0,\infty ;L^2(\omega ))\) follows from the convergence in \(L^2(Q_\omega )\) and the fact that \(\Vert \hat{u}_T\Vert _{L^\infty (0,\infty ;L^2(\omega ))} \le \gamma \) for every \(T > 0\). \(\square \)
In the previous theorem we proved the existence of local minimizers \(\{u_T\}_{T > T_0}\) of problems \(({\hbox {P}}_{T}) \)converging to \(\bar{u}\) assuming that \(\bar{u}\) is a strict local minimizer of (P). Moreover, in the proof of the theorem, the existence of an \(L^2(Q_\omega )\)-closed ball \(B_\rho (\bar{u})\) such that the minimum of \(J_T\) on the set \({U_{ad}}\cap B_\rho (\bar{u})\) is achieved at the local minimizer \(u_T\) was established. In particular, this implies that \(J_T(u_T) \le J_T(\bar{u})\) for every \(T > T_0\). In the next theorem the following question is addressed: if \(\{u_T\}_{T > T_0}\) is a sequence of local minimizers of problems \(({\hbox {P}}_{T}) \)converging to \(\bar{u}\), does the inequality \(J_T(u_T) \le J_T(\bar{u})\) hold for T large enough? The positive answer to this question is also important to establish the estimates in Theorem 4.4 below.
Theorem 4.3
Suppose that \({U_{ad}}\) is defined by (1.2) or (1.3). Let \(\bar{u}\) be a local minimizer of (P) satisfying the second order sufficient optimality condition given in Theorems 3.3 and 3.5, respectively. Let \(\{u_T\}_{T > T_0}\) be a sequence of local minimizers of problems \(({\hbox {P}}_{T}) \)such that \(\hat{u}_T \rightarrow \bar{u}\) strongly in \(L^2(Q_\omega )\). Then, there exists \(T_0^* \in (T_0,\infty )\) such that \(J_T(u_T) \le J_T(\bar{u})\) for every for every \(T \ge T^*_0\).
Proof
The proof is carried out under the assumption that \({U_{ad}}\) is given by (1.2). It is similar, even easier, if \({U_{ad}}\) is given by (1.3). First, we observe that the convergence \(\hat{u}_T \rightarrow \bar{u}\) in \(L^2(Q_\omega )\) and the fact that \(\Vert \hat{u}_T(t)\Vert _{L^2(\omega )} \le \gamma \) for almost every \(t > 0\) implies that \(\hat{u}_T \rightarrow \bar{u}\) strongly in \(L^p(0,\infty ;L^2(\omega ))\) for every \(p < \infty \). Then, for fixed \(p > \frac{4}{4 - n}\), there exists \(\hat{T} \ge T_0\) such that \(\hat{u}_T \in \mathcal {U}_p\) for every \(T \ge \hat{T}\). This yields \(\hat{y}_T = G_p(\hat{u}_T) \rightarrow G_p(\bar{u}) = \bar{y}\) in \(Y_p\) as \(T \rightarrow \infty \). Given the adjoint state \(\varphi _T\) associated with \(u_T\), we denote by \(\hat{\varphi }_T\) its extension by 0 for \(t > T\).
We proceed by contradiction. If the statement fails, then there exists a sequence \(\{u_{T_k}\}_{k = 1}^\infty \) with \(T_k \rightarrow \infty \) as \(k \rightarrow \infty \) such that
Let us set \(\rho _k = \Vert \hat{u}_{T_k} - \bar{u}\Vert _{L^2(Q_\omega )}\) and \(v_{T_k} = \frac{1}{\rho _k}(\hat{u}_{T_k} - \bar{u})\). Taking a subsequence, denoted in the same way, we have \(v_{T_k} \rightharpoonup v\) in \(L^2(Q_\omega )\).
Now, we split the proof in three steps.
Step I \(\hat{\varphi }_T \rightarrow \bar{\varphi }\) in \(W(0,\infty ) \cap L^\infty (Q)\) as \(T \rightarrow \infty \). Let us set \(\psi _T = \hat{\varphi }_T - \bar{\varphi }\) and denote by \(\chi _T\) the real function taking the value 1 if \(t \in [0,T]\) and 0 otherwise. Then, \(\psi _T\) satisfies the equation
Since \(\hat{y}_T \rightarrow \bar{y}\) in \(Y_p\), we deduce that \(\hat{y}_T \rightarrow \bar{y}\) in \(L^q(0,\infty ;L^2(\varOmega )) \cap L^\infty (Q)\) for every \(q \ge 2\). Hence, with the mean value theorem and (3.2) we obtain that \(\Big [\frac{\partial f}{\partial y}(x,t,\bar{y}) - \frac{\partial f}{\partial y}(x,t,\hat{y}_T)\Big ] \rightarrow 0\) in \(L^q(0,\infty ;L^2(\varOmega ))\). Moreover, from Theorem A.4 and the fact that \(y_d \in L^2(Q) \cap L^p(0,\infty ;L^2(\varOmega ))\), we get that \(\hat{\varphi }_T\) is bounded in \(W(0,\infty ) \cap L^\infty (Q)\). Therefore the first term of the right hand side in the above partial differential equation converges to 0 in \(L^q(0,\infty ;L^2(\varOmega ))\). The same convergence is true for the second term \(\chi _T(\hat{y}_T - \bar{y})\). The third term \((1- \chi _T)(\bar{y} - y_d)\) converges to 0 in \(L^q(0,\infty ;L^2(\varOmega ))\) for \(q = p\) if \(p < \infty \) and \(q < \infty \) arbitrary if \(p = \infty \). Then, from Theorem A.4 the claimed convergence \(\hat{\varphi }_T \rightarrow \bar{\varphi }\) in \(W(0,\infty ) \cap L^\infty (Q)\) follows.
Step II \(v \in C_{\bar{u}}\). Using the local optimality of \(\bar{u}\) we get
On the other side, using the convergence established in Step I and the convergence \(\hat{u}_{T_k} \rightarrow \bar{u}\) in \(L^2(Q_\omega )\) along with the local optimality of \(u_{T_k}\) we infer
The last two inequalities imply that \(J'(\bar{u})v = 0\). Now, the proof continues as in the Step I of the proof of Theorem 3.3.
Step III—Contradiction Since \((u_{T_k},\varphi _{T_k})\) satisfies (2.10), we deduce the inequality \(\mathcal {L}_{T_k}(\bar{u},\lambda _{T_k}) < \mathcal {L}_{T_k}(u_{T_k},\lambda _{T_k})\) with (4.3) and the fact that \(\lambda _{T_k}(t)\Vert \bar{u}(t)\Vert _{L^2(\omega )} \le \lambda _{T_k}(t)\gamma = \lambda _{T_k}(t)\Vert u_{T_k}(t)\Vert _{L^2(\omega )}\). Hence, performing a Taylor expansion and using (4.2) we infer
Dividing the above expression by \(\rho _k^2/2\) we get
We observe that for \(k \rightarrow \infty \)
Setting \(u_{\theta _k} = \bar{u} + \theta _k(u_{T_k} - \bar{u})\), we denote by \(y_{\theta _k}\) the solution of (4.1) corresponding to the control \(u_{\theta _k}\) and by \(\varphi _{\theta _k}\) the corresponding adjoint state in \(Q_{T_k,\omega }\). Then, putting \(\psi _k = \hat{\varphi }_{\theta _k} - \bar{\varphi }\) we have
Arguing as in Step I we obtain that \(\psi _k \rightarrow 0\) in \(W(0,\infty ) \cap L^\infty (Q)\). Then, arguing as in Steps II and III of the proof of Theorem 3.3 and using the established convergences, we infer that \(\frac{\partial ^2\mathcal {L}}{\partial u^2}(\bar{u},\bar{\lambda })v^2 \le 0\) and the contradiction follows. \(\square \)
Under an extra assumption on f, the following theorem provides estimates for the difference \(\hat{u}_T - \bar{u}\).
Theorem 4.4
Suppose that \({U_{ad}}\) is defined by (1.2) or (1.3) and that \(\bar{u}\) is a local minimizer of (P) satisfying the second order sufficient optimality condition. We assume that \(\frac{\partial f}{\partial y}(x,t,y) \ge 0\) holds for all \(y \in {\mathbb {R}}\) and almost all \((x,t) \in Q\). Let \(\{u_T\}_{T > T_0}\) be a sequence of local minimizers of problems \(({\hbox {P}}_{T}) \)such that \(\hat{u}_T \rightarrow \bar{u}\) in \(L^2(Q_\omega )\). Then, there exist \(T^* \in [T_0,\infty )\) and a constant C such that for every \(T \ge T^*\)
Proof
We use the inequalities (3.9) or (3.17). For this purpose, we take \(T^* \in [T_0^*,\infty )\) such that \(\Vert \hat{u}_T - \bar{u}\Vert _{L^2(Q_\omega )} < \varepsilon \) for all \(T \ge T^*\), where \(T_0^*\) is introduced in Theorem 4.3. Then, given \(T \ge T^*\), (3.9) or (3.17), and Theorem 4.3 yield
which leads to
To prove the first estimate of (4.4) we observe that \(\hat{y}_T\) satisfies the equation
Testing this equation with \(\hat{y}_T\), and using that \(f(x,t,\hat{y}_T)\hat{y}_T \ge 0\) due to the monotonicity of f with respect to y and (1.4), it follows that
From this inequality we infer with (1.9) that
This inequality and (4.5) imply the estimate of the controls in (4.4). To get the estimate for the states we observe that \(\phi _T = \hat{y}_T - \bar{y}\) satisfies the equation
where \(y_{T,\theta } = \bar{y} + \theta _T(\hat{y}_T - \bar{y})\) with \(\theta _T:Q \longrightarrow [0,1]\) measurable. Then, applying Theorem A.3 and Remark 5.2 we infer \(\Vert \phi _T\Vert _{W(0,\infty )} \le K_3\Vert \hat{u}_T - \bar{u}\Vert _{L^2(Q_\omega )}\). Combining this estimate with the one established for the controls we deduced (4.4). \(\square \)
References
Aseev, S.M., Krastanov, M.I., Veliov, V.M.: Optimality conditions for discrete-time optimal control on infinite horizon. Pure Appl. Funct. Anal. 2(3), 395–409 (2017)
Aseev, S.M., Kryazhimskiy, A.V.: The Pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons. SIAM J. Control Optim. 43(3), 1094–1119 (2004)
Azmi, B., Kunisch, K., Rodrigues, S.: Saturated feedback stabilizability to trajectories for the Schlögl parabolic equation. Technical report, arXiv:2111.01329 [math.OC], (2021)
Basco, V., Cannarsa, P., Frankowska, H.: Necessary conditions for infinite horizon optimal control problems with state constraints. Math. Control Relat. Fields 8(3–4), 535–555 (2018)
Bonnans, F.: Second-order analysis for control constrained optimal control problems of semilinear elliptic systems. Appl. Math. Optim. 38, 305–325 (1998)
Brezis, H.: Functional Analysis. Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)
Carlson, D.A., Haurie, A.B., Leizarowitz, A.: Infinite Horizon Optimal Control. Deterministic and Stochastic Systems. Springer, Berlin (1991). Second revised and enlarged edition of the (1987) original
Casas, E., Kunisch, K.: Stabilization by sparse controls for a class of semilinear parabolic equations. SIAM J. Control Optim. 55(1), 512–532 (2017)
Casas, E., Kunisch, K.: Infinite horizon optimal control problems for a class of semilinear parabolic equations. SIAM J. Control Optim. 60(4), 2070–2094 (2022)
Casas, E., Kunisch, K.: Optimal control of semilinear parabolic equations with non-smooth pointwise-integral control constraints in time-space. Appl. Math. Optim. 85(1), 12 (2022)
Casas, E., Kunisch, K.: Infinite horizon optimal control problems with discount factor on the state. Part I: analysis of the controlled state equation. SIAM J. Control Optim. to appear (2023)
Casas, E., Kunisch, K.: Infinite horizon optimal control problems with discount factor on the state. Analysis of the control problem. SIAM J. Control Optim. to appear, Part II (2023)
Casas, E., Tröltzsch, F.: Second order analysis for optimal control problems: improving results expected from from abstract theory. SIAM J. Optim. 22(1), 261–279 (2012)
Gunzburger, M.D., Manservisi, S.: The velocity tracking problem for Navier-Stokes flows with bounded distributed controls. SIAM J. Control Optim. 37(6), 1913–1945 (1999)
Halkin, H.: Necessary conditions for optimal control problems with infinite horizons. Econometrica 42(2), 267–272 (1974)
Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence (1988)
Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Math. Surv. and Monogr, vol. 49. American Mathematical Society, Providence (1997)
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Eduardo Casas was supported by MCIN/ AEI/10.13039/501100011033/ under research Project PID2020-114837GB-I00.
Appendix
Appendix
Here we prove \(L^\infty (Q)\) estimates for the solution of the following equation
assuming that \(y_0 \in L^\infty (\varOmega )\) and \(g \in L^2(Q) \cap L^p(0,\infty ;L^2(\varOmega ))\) with p satisfying (1.10).
Definition A.1
We call y a solution to (A.1) if \(y \in L_{loc}^2(0,\infty ;H^1(\varOmega ))\), and for every \(T > 0\) the restriction of y to \(Q_T = \varOmega \times (0,T)\) belongs to \(W(0,T) \cap L^\infty (Q_T)\) and satisfies the following equation in the variational sense
Since \(y \in L^\infty (Q_T)\), we observe that (1.8) implies for \(M_T = \Vert y\Vert _{L^\infty (Q_T)}\) and for almost all \((x,t) \in Q_T\)
Theorem A.2
Under the assumption (1.4)–(1.6), equation (A.1) has a unique solution y. In addition, if \(y \in L^2(Q)\), then \(y \in W(0,\infty ) \cap L^\infty (Q)\) and \(f(\cdot ,\cdot ,y) \in L^2(Q) \cap L^\infty (Q)\) holds. Moreover, the following estimates are satisfied
where \(M_f\) is given by (1.5), \(K_\infty = \Vert y\Vert _{L^\infty (Q)}\), \(C_{K_\infty }\) as in (1.6) with \(M = K_\infty \), and
Proof
The existence and uniqueness of a solution y of (A.1) is a consequence of [10, Theorem 2.1]. Now, we assume that \(y \in L^2(Q)\). The proof is split into several steps.
Step I \(y \in L^2(0,\infty ;H^1(\varOmega )) \cap L^\infty (0,\infty ;L^2(\varOmega ))\). Testing equation (A.2) with y, integrating in (0, t) with \(t \in (0,T)\), using (1.5), and arguing as in (A.3) we get
where \(\varOmega _f(t) = \{x \in \varOmega : |y(x,t)| < M_f\}\). This inequality along with (1.9) proves that \(y \in L^2(0,\infty ;H^1(\varOmega )) \cap L^\infty (0,\infty ;L^2(\varOmega ))\) and (A.4) holds.
Step II \(y \in L^\infty (Q)\). Let us first observe that without loss of generality we may suppose that \(y_0 \in H^1(\varOmega )\). Indeed, if this is not the case we use the fact that \(f(\cdot ,\cdot ,y) \in L^2(Q_T)\) by (A.3) and \(g \in L^2(Q_T)\) to deduce that \(y \in C([T_0,T];H^1(\varOmega ))\) for each \(0< T_0< T < \infty \); see, for instance, [17, Corollary III.2.4]. Since \(y \in L^\infty (Q_{T_0})\) for each \(0< T_0 < \infty \), it is enough to prove that \(y \in L^\infty (\varOmega \times (T_0,\infty ))\). Then there is no loss of generality if we assume that \(y_0 \in H^1(\varOmega )\) and, consequently, \(y \in H^1(Q_T)\) for every \(T < \infty \); see [17, Proposition III.2.5].
For every real number \(\rho \ge \max \{\Vert y_0\Vert _{L^\infty (\varOmega )},M_f\}\), \(M_f\) given by (1.5), we introduce the function \(y_\rho (x,t) = y(x,t) - {\text {Proj}}_{[-\rho ,+\rho ]}(y(x,t))\). Then, we still have that \(y_\rho \in H^1(Q_T)\) for all \(T < \infty \). We set \(A_\rho (t) = \{x \in \varOmega : |y(x,t)| > \rho \}\) for every \(t \in (0,\infty )\).
First we prove the result for \(n = 2\) or 3. Let us choose a number \(\alpha \in \big (\frac{pn}{2p-4},\frac{n}{n-2}\big )\). Observe that \(\frac{pn}{2p-4} < \frac{n}{n-2}\) obviously holds if \(n = 2\) and it is also true for \(n = 3\) due to the assumption \(p > \frac{4}{4 - n}\). Note that \(\alpha \) satisfies \(\alpha \in (1,\infty )\) in case \(n=2\), and \(\alpha \in (\frac{3}{2},3)\) in case \(n=3\). We will denote \(\alpha ' = \frac{\alpha }{\alpha - 1}\).
Testing equation (A.2) with \(y_\rho \), integrating in (0, t) with \(t \in (0,T)\), and using that \(\frac{\partial y}{\partial t}y_\rho = \frac{\partial y_\rho }{\partial t}y_\rho \), \(\nabla y\cdot \nabla y_\rho = |\nabla y_\rho |^2\), and \(f(x,t,y)y_\rho \ge 0\) due to (1.5) and \(\rho \ge M_f\), we infer
where \(C_a\) is given by (1.9) and \(C_1\) is the embedding constant of \(H^1(\varOmega ) \subset L^{2\alpha }(\varOmega )\). The above estimate leads to
From the above inequalities we get
with \(C_2\) independent of T and \(\rho \).
On the other hand, taking \(\kappa = 1 + \frac{2\alpha '(p - 2)}{pn}\), \(r = \frac{2p\kappa }{\alpha '(p-2)}\), and \(q = 2\kappa \) we get \(\frac{1}{r} + \frac{n}{2q} = \frac{n}{4}\). Then, using [16, Formula II-(3.4)] we obtain the existence of a constant \(C_3\) independent of T such that
For every \(j = 0, 1, 2, \ldots \) we set \(k_j = \rho (2 - 2^{-j})\). We observe that \(\rho \le k_j \le 2\rho \), \(A_\rho (t) = A_{k_0}(t) \supset A_{k_1}(t) \supset A_{k_2}(t) \supset \ldots \), and \(A_{k_j}(t) \supset A_{2\rho }(t)\) for every \(j \ge 0\). Then, we have
Combining this inequality with (A.8) and (A.9) we deduce
Using that \(k_{j + 1} - k_j = \rho 2^{-(j+1)}\) we infer
Setting
for \(j \ge 0\), we have \(\xi _{j + 1} \le cb^j\xi _j^{\frac{\kappa }{\alpha '}}\) for \(j \ge 0\). Moreover, since \(\alpha > \frac{pn}{2p - 4}\) we get that \(\beta = \frac{\kappa }{\alpha '} > 1\). Then, from [16, LemmaII\(-\)5.6] we obtain
Let us estimate \(\xi _0\). For this purpose we distinguish two cases. First, we assume that \(n = 2\) and \(p \in (2,4]\). Using that \(|A_\rho (t)| \le \frac{1}{\rho ^2}\Vert y(t)\Vert ^2_{L^2(A_\rho (t)}\), we get
The last inequality follows from the fact \(y \in L^2(Q) \cap L^\infty (0,\infty ;L^2(\varOmega ))\) and \(\frac{2p}{\alpha '(p - 2)} \ge 2\) because \(\alpha ' \in (1,\frac{p}{2})\) and \(\frac{p}{2} \le \frac{p}{p - 2}\) for \(p \le 4\). For the remaining cases we observe that \(\frac{2\alpha '(p-2)}{p} > 1\) and additionally \(\frac{2\alpha '(p-2)}{p} < 6\) if \(n = 3\). Now, we argue as follows
Selecting
we get with (A.10)
as \(j \rightarrow \infty \). Finally, we get
Hence, \(|A_{2\rho }(t)| = 0\) for almost every \(t \in (0,T)\) holds. Since \(T > 0\) was arbitrarily selected and all the constants above are independent of T, we deduce that \(|y(x,t)| \le 2\rho \) for almost all \((x,t) \in Q\) and (A.5) follows with (A.4).
Now, we explain the changes in the proof for the case \(n = 1\). To get an analogous inequality to (A.8), we use the following Gagliardo-Nirenberg inequality
see, for instance, [6, P. 233]. Then, we have with Hölder and Young inequalities
From here we infer
On the other side, we apply (A.9) with \(r = 8\) and \(q = 4\) and arguing as for the cases \(n = 2\) or 3 we obtain
Thus we get
To estimate \(\xi _0\) we use again that \(y \in L^2(Q) \cap L^\infty (0,\infty ;L^2(\varOmega ))\) and proceed as follows
The rest of the proof follows as for the cases \(n = 2\) or 3.
Step III—Proof of (A.6) and (A.7). The inequalities of (A.6) are an immediate consequence of (1.4), (1.6), and the mean value theorem:
with \(0 \le \theta (x,t) \le 1\).
Since \(y \in L^2(0,\infty ;H^1(\varOmega ))\), we have that \(\varDelta y \in L^2(0,\infty ;H^1(\varOmega )^*)\). From the state equation and \(g, f(\cdot ,\cdot ,y) \in L^2(Q)\) we infer that \(\frac{\partial y}{\partial t} \in L^2(0,\infty ;H^1(\varOmega )^*)\) and, hence, \(y \in W(0,\infty )\). Finally, the fact that \(y \in W(0,\infty )\) implies (A.7); see [9, Theorem 2.4] for details. \(\square \)
Remark 5.1
The proof of the boundedness of y in Q follows some ideas of the proof of [16, Theorem III\(-\)7.1]. In that theorem, the boundedness is established for finite time horizon and the \(L^\infty (Q_T)\) estimates depend on time T. In our theorem, we have avoided the dependence with respect to time exploiting the fact that \(y \in L^2(Q)\), which was used to estimate \(\xi _0\). By a simple modification of our proof, the \(L^\infty (Q)\) estimate of y can be also obtained in terms of \(\Vert g\Vert _{L^r(0,\infty ;L^q(\varOmega ))}\) if \(\frac{1}{r} + \frac{n}{2q} < 1\). We observe that the assumption \(y \in L^2(Q)\) is natural in the context of our optimal control problem due to the structure of its cost functional. Another difference of our estimates with respect to [16, Theorem III\(-\)7.1] concerns the choice of the boundary condition. Here we have treated the Neumann case while the Dirichlet case was considered in the mentioned reference. The only difference in our proof for the Dirichlet case consists in the definition of \(\rho \) that should include the \(L^\infty (\varSigma )\) norm of the Dirichlet datum, if it is not zero.
Now, we analyze the following linear equation
We assume that
and that \(d:Q \times {\mathbb {R}} \longrightarrow {\mathbb {R}}\) is a Carathéodory function satisfying
for almost all \((x,t) \in Q\).
Theorem A.3
Under assumptions (A.13)–(A.15), equation (A.12) has a unique solution \(z \in W(0,\infty )\) for every \(h \in L^2(0,\infty ;H^1(\varOmega )^*)\) and \(z_0 \in L^2(\varOmega )\), which satisfies
where \(K_{3,y}\) depends on \(\Vert y\Vert _{L^\infty (Q)}\). In addition, if \(h \in L^2(Q) \cap L^p(0,\infty ;L^2(\varOmega ))\) with \(p \in (\frac{4}{4 - n},\infty ]\) and \(z_0 \in L^\infty (\varOmega )\), then \(z \in L^\infty (Q)\) and the estimate
holds for a constant \(K_{4,y}\) also depending on \(\Vert y\Vert _{L^\infty (Q)}\).
Proof
Due to the fact that \(y \in L^\infty (Q)\) and (A.14) we have that \(d(\cdot ,\cdot ,y) \in L^\infty (Q)\), and hence the existence and uniqueness of \(z \in W(0,T) \cap L^\infty (Q_T)\) holds for every \(T < \infty \). Let us prove that \(z \in W(0,\infty )\). We put \(K = \Vert y\Vert _{L^\infty (Q)}\) and \(C_K = \Vert d(\cdot ,\cdot ,y)\Vert _{L^\infty (Q)}\). Given \(\delta _d \in [0,1)\) we know that there exists a constant \(C_{a,\delta _d}\) such that
We select \(\varepsilon > 0\) such that \(\max \{C_d,\frac{C_K}{m_d}\}C_1^2\varepsilon \le \frac{C^2_{a,\delta _d}}{4}\), where \(m_d\) and \(C_d\) are given in (A.15) and \(C_1\) is the embedding constant for \(H^1(\varOmega ) \subset L^4(\varOmega )\). Using (A.13) we deduce the existence of \(T_\varepsilon > 0\) such that
For \(t > 0\) we set \(\varOmega _{m_d}(t) = \{x \in \varOmega : |y(x,t)| \le m_d\}\). Now, we test (A.12) with z and integrate over \(\varOmega \times (T_\varepsilon ,t)\) for every \(t > T_\varepsilon \), use assumption (A.15), and (A.19)
This implies
Since z solves (A.12) in \((0,T_\varepsilon )\), we have \(z \in W(0,T_\varepsilon )\) and \(\Vert z\Vert _{W(0,T_\varepsilon )}\) can be estimated by \(\Vert h\Vert _{L^2(0,\infty ;H^1(\varOmega )^*)} + \Vert z_0\Vert _{L^2(\varOmega )}\). This along with the above estimate implies the desired estimate of z in \(L^2(0,\infty ;H^1(\varOmega )) \cap L^\infty (0,\infty ;L^2(\varOmega ))\). From the equation (A.12) we infer that \(\frac{\partial z}{\partial t} \in L^2(0,\infty ;H^1(\varOmega )^*)\) and estimate (A.16) follows.
Finally, under the additional regularity of h and \(z_0\), applying Theorem A.2 to the equation
with \(f = 0\) and \(M_f = 0\) there, we infer that \(z \in L^\infty (Q)\) and (A.17) holds. Here we have used that \(L^\infty (0,\infty ;L^2(\varOmega )) \cap L^2(Q) \subset L^p(0,\infty ;L^2(\varOmega ))\) for every \(p \ge 2\). \(\square \)
We finish this appendix by analyzing the following adjoint equation
Theorem A.4
Under assumptions (A.13)–(A.15), equation (A.20) has a unique solution \(\varphi \in W(0,\infty )\) for all \(h \in L^2(0,\infty ;H^1(\varOmega )^*)\) which satisfies
where \(K_{5,y}\) depends on \(\Vert y\Vert _{L^\infty (Q)}\). In addition, if \(h \in L^2(Q) \cap L^p(0,\infty ;L^2(\varOmega ))\) with \(p \in (\frac{4}{4 - n},\infty ]\), then \(\varphi \in L^\infty (Q)\) and the estimate
holds for a constant \(K_{6,y}\) also depending on \(\Vert y\Vert _{L^\infty (Q)}\).
Proof
First, we prove uniqueness. For this purpose we establish that the unique solution to (A.20) with \(h = 0\) is \(\varphi = 0\). Indeed, in this case, we take \(z \in W(0,\infty )\) solution to (A.12) with \(h = \varphi \) and \(z_0 = 0\). Then we have,
To prove the existence of a solution we denote by \(\varphi _T \in W(0,T)\) the solutions of
The existence and uniqueness of \(\varphi _T\) is known because the function d(x, t, y(x, t)) is bounded. We extend \(\varphi _T\) by 0 to \((0,\infty )\) and estimate this extension in \(L^2(0,\infty ;H^1(\varOmega ))\) independently of T. For this purpose we take \(\phi \in L^2(0,\infty ;H^1(\varOmega )^*)\) arbitrary and denote by z the solution of (A.12) with \(h = \phi \) and \(z_0 = 0\). Then, we have with (A.16)
This implies that
From (A.23) and the above estimate we deduce the boundedness of \(\{\varphi _T\}_{T > 0}\) in \(W(0,\infty )\). Then, there exists a sequence \(\{T_k\}_{k = 1}^\infty \) with \(T_k \rightarrow \infty \) and a function \(\varphi \in W(0,\infty )\) such that \(\varphi _{T_k} \rightharpoonup \varphi \) in \(W(0,\infty )\) as \(k \rightarrow \infty \). It is obvious that we can pass to the limit in (A.23) and deduce that \(\varphi \) satisfies (A.20) and estimate (A.21) holds.
To prove that \(\varphi \in L^\infty (Q)\) under the additional regularity assumption on h we introduce the functions \(z_T(x,t) = \varphi _T(x,T-t)\) for every \(T > 0\). Then \(z_T \in W(0,T)\) and it satisfies (A.12) in \(Q_T\) with \(z_0 = 0\) and \(h_T(x,t) = h(x,T-t)\). Since
\(\Vert h_T\Vert _{L^2(Q_T)} = \Vert h\Vert _{L^2(Q_T)} \le \Vert h\Vert _{L^2(Q)}\) and \(\Vert h_T\Vert _{L^p(0,\infty ;L^2(\varOmega ))} \le \Vert h\Vert _{L^p(0,\infty ;L^2(\varOmega ))}\), we infer from Theorem A.2 that \(\{\varphi _T\}_{T > 0}\) is uniformly bounded in \(L^\infty (Q)\) and, consequently, estimate (A.22) holds. \(\square \)
Remark 5.2
If the function f in (A.1) satisfies \(\frac{\partial f}{\partial y}(x,t,y) \ge 0\) for every \(y \in {\mathbb {R}}\) and almost all \((x,t) \in Q\), then the term \(\Vert y\Vert _{L^2(Q)}\) in the estimates (A.4) and (A.5) can be removed. Under this assumption on f, the constants \(M_f\) and \(C_{M_f}\) in (1.5) and (1.7) are zero. Then, it is enough to use this in the proof of Theorem A.2 to get the independence of the estimates with respect to y.
Moreover, by an analogous argument, if the assumptions (A.13)–(A.15) are replaced by
and almost all \((x,t) \in Q\), then the constants \(K_{3,y}\) until \(K_{6,y}\) in the estimates (A.16), (A.17), (A.21), and (A.22) can be chosen independently of y.
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Casas, E., Kunisch, K. Infinite Horizon Optimal Control for a General Class of Semilinear Parabolic Equations. Appl Math Optim 88, 47 (2023). https://doi.org/10.1007/s00245-023-10006-4
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DOI: https://doi.org/10.1007/s00245-023-10006-4
Keywords
- Infinite horizon optimal control
- Semilinear parabolic equations
- First and second order optimality conditions
- \(L^\infty \) regularity