Optimal control problems for a von Kármán system with long memory

In this paper, we study quadratic cost optimal control problems governed by a von Kármán system with long memory. We prove the existence of an optimal control for the cost. Then, by proving the strong Gâteaux differentiability of nonlinear solution mapping we establish necessary optimality condition for the optimal control corresponding to the quadratic cost. Further, we study the time local uniqueness of the optimal controls for distributive observation.

We consider a von Kármán plate model with internal damping and long memory. In the context of control theory, early results for the von Kármán plate can be found in [1], which gives the derivation of the model and asymptotic energy estimates for the system.

In this paper, our system may be described as follows: Let Ω be an open bounded domain in \(R^{2}\) with a sufficiently smooth boundary ∂Ω. In \((0, T) \times\Omega\), we consider the following von Kármán system with long memory and the clamped boundary condition in the variables y, representing the position of the plate and the Airy’s stress function v:

where the vector ν denotes an outward normal, \(k \in C^{1}([0,T])\) is a memory kernel, f is a forcing function, and the von Kármán bracket is given by

The aim of this paper can be summarized as follows.

Firstly, we survey the well-posedness of Eq. (1.1) with respect to y in the Hadamard sense relying on some previous results. To name just a few, we can refer to [2–4], and references therein. Especially, in order to prove the local Lipschitz continuity of the solution mapping, we employ the energy equality of Volterra-type integro-differential equation which is proved in [5].

Secondly, based on this result, we study the following optimal control problem:

subject to

where B is a controller, u is a control, J is a quadratic cost function, \(y(u)\) denotes the state for a given \(u \in{ \mathcal {U}}\), and \({\mathcal {U}}\) is a Hilbert space of control variables. In order to apply the variational approach due to Lions [6] to our problem, we propose the quadratic cost functional J as studied in Lions [6], which is to be minimized within \({\mathcal {U}}_{\mathrm{ad}}\), an admissible set of control variables in \({\mathcal {U}}\).

The quadratic cost optimal control problem consists of two problems, to show the existence of optimal control and to derive a necessary condition for the optimal control.

We need to show the existence of \(u^{*} \in {\mathcal {U}}_{\mathrm{ad}} \) that minimizes the quadratic cost functional J. However, differently from the linear equation case, we are faced with difficulty that the weak convergence of the controlled state \(y(u_{n} )\) is insufficient to cover the convergence of the nonlinear part of Eq. (1.3). Therefore, it is necessary to improve the convergence of the controlled state \(y(u_{n} )\). Thus, to improve the convergence, we newly adapt the idea of Dautray and Lions ([7], pp.578-581), namely, the strong convergence result studied in linear evolution equations. Also, this method is quite useful in proving the strong Gâteaux differentiability of the nonlinear solution mapping \(u \rightarrow y(u)\), which is used to define the associate adjoint system. Then, we establish a necessary condition of optimality of the optimal control \(u^{*}\) for some physically meaningful observation case employing the associate adjoint system.

In author’s knowledge, this is a newly developed method.

In fact, the extension of optimal control theory to quasilinear equations is not easy. Only few researches have been devoted to the study of optimal control or identification problems in specific quasilinear equations. For instance, we can refer to Hwang and Nakagiri [8, 9] and Hwang [10, 11].

Moreover, in this paper, we discuss the time local uniqueness of optimal control for distributive observation. As is widely known, it is unclear and difficult to verify the uniqueness of optimal control in nonlinear control problems.

Following the idea in [12], we show the strict convexity of the quadratic cost J for distributive observation in local time interval by making use of the second-order Gâteaux differentiability of the nonlinear solution mapping \(u \rightarrow y(u)\). As a consequence, we prove the time local uniqueness of optimal control. This is another novelty of the paper.

If X is a Banach space, we denote by \(X'\) its topological dual and by \(\langle\cdot, \cdot \rangle_{X', X}\) the duality pairing between \(X'\) and X. We introduce the following abbreviations:

with \(p \ge1\), where \(W^{k, p}\) is the \(L^{p} \)-based Sobolev space. When \(p=2\), the space becomes a Hilbert space, and we use the special notation \(H^{k}\) to denote \(W^{k, 2}\) for \(k \ge1\), and \(H^{k}_{0}\) mean the completions of \(C^{\infty}_{0}(\Omega)\) in \(H^{k}\) for \(k \ge1\).

We denote the scalar product on \(L^{2}\) by \((\cdot,\cdot)_{2}\). Then the scalar products on \(H^{k}_{0}\) (\(k=1,2\)) are given as follows:

Then obviously,

and \(D(\Delta^{2} ) = H^{4} \cap H^{2}_{0}\).

Especially, the duality pairs between \(H^{k}_{0}\) and \(H^{-k}\) (\(k =1,2\)) are abbreviated by \(\langle\cdot, \cdot\rangle_{k, -k}\). It is clear that \(H^{2}_{0} \hookrightarrow H^{1}_{0} \hookrightarrow L^{2} \hookrightarrow H^{-1} \hookrightarrow H^{-2}\), each space is dense in the next one, and the injections are continuous.

It is well known that the biharmonic operator

is bijective and admits an isometric extension

Thus, we can define the operator \(G \in {\mathcal {L}}(L^{2}, H^{4} \cap H^{2}_{0})\) (or \({\mathcal {L}}(H^{-2}, H^{2}_{0})\)) by

Therefore, from Eq. (1.1) we can also note that

We further collect some results for the Airy stress function and von Kármán bracket.

Lemma 2.1

The trilinear form \(b: H^{2}_{0} \times H^{2}_{0} \times H^{2}_{0} \to R\) given by

satisfies the property

Proof

See [13]. □

Lemma 2.2

1. (1)

   [3, 14] The bilinear forms \((\psi, \phi) \to G [\psi, \phi] \) from \(H^{2} \times H^{2}\) into \(W^{2, \infty}\) and \((\psi, \phi) \to [\psi, \phi] \) from \(H^{1} \times H^{2}\) into \(H^{-2}\) are continuous. We also have the following estimates:

   $$\begin{aligned}& \bigl\Vert G [ \psi, \phi] \bigr\Vert _{W^{2, \infty}} \le C \Vert \psi \Vert _{H^{2}} \Vert \phi \Vert _{H^{2}},\quad \psi, \phi\in H^{2}, \end{aligned}$$

   (2.3)
   $$\begin{aligned}& \bigl\Vert [\psi, \phi] \bigr\Vert _{H^{-2} } \le C \Vert \psi \Vert _{H^{1}} \Vert \phi \Vert _{H^{2}},\quad \psi \in H^{1}, \phi\in H^{2}. \end{aligned}$$

   (2.4)

   Consequently,

   $$ \bigl\Vert \bigl[\varphi, G[\psi, \phi] \bigr] \bigr\Vert \le C \Vert \varphi \Vert _{H^{2}}\Vert \psi \Vert _{H^{2}} \Vert \phi \Vert _{H^{2}},\quad \varphi, \psi, \phi\in H^{2}. $$

   (2.5)
2. (2)

   [2], Lemma 3.2. The bilinear form \([\cdot, \cdot ]: H^{2}_{0} \times H^{2}_{0} \to H^{-1-\epsilon} \) given by

   $$(\psi, \phi) \to[\psi, \phi] $$

   is continuous for every \(\epsilon> 0\). Moreover,

   $$\bigl\Vert [\psi, \phi] \bigr\Vert _{H^{-1-\epsilon}} \le C \Vert \psi \Vert _{H^{2}_{0} } \Vert \phi \Vert _{H^{2}_{0}}. $$

The solution Hilbert space \(W(0, T)\) of Eq. (1.1) is defined by

endowed with the norm

Definition 3.1

We say that a function y is a weak solution of Eq. (1.1) if \(y \in W(0, T)\) and satisfies

In the sequel, we give the important energy equality of weak solutions of Eq. (1.1). However, we are faced with the difficulty of regularity of weak solutions of Eq. (1.1), that is, \(y' \) generally does not belong to \(H^{2}_{0}\) as notified before. In order to overcome this difficulty, we employ the idea of Lions and Magenes [15], pp.276-279, namely, double regularization method used in linear hyperbolic equations. We also note that the method has been applied in [5], Proposition 2.1, to study a semilinear second-order integro-differential equation.

Lemma 3.1

Let X, Y be two Banach spaces, \(X \subset Y\) densely, and X be reflexive. Set

Then

Proof

See [15], p.275. □

Lemma 3.2

Assume that y is a weak solution of Eq. (1.1). Then we can assert (after possibly a modification on a set of measure zero) that

Proof

Assume that y is a weak solution of Eq. (1.1). Then by referring to the results as in [2] (cf. [4]) we have

From the inclusion \(W(0, T) \subset C([0, T]; H^{1}_{0} ) \cap C^{1} ( [0, T]; L^{2})\) (see [7]) and also from \(C([0, T]; H^{k}_{0}) \subset C_{s}([0, T]; H^{k}_{0} )\) (\(k=1,2\)) we can obtain by (3.3) that

Thus, by Lemma 3.1 we have (3.2). □

Proposition 3.1

Assume that y is a weak solution of Eq. (1.1). Then, for each \(t \in[0, T]\), we have the energy equality

where \(\Delta v_{0} = - \Delta^{-1}[y_{0}, y_{0}]\).

Proof

By Lemma 3.2 and the uniform boundedness theorem, we have \(y(t) \in H^{2}_{0} \) and \(y'(t) \in H^{1}_{0} \) for all \(t \in[0, T]\). Thus, all functions in (3.4) have meaning for all \(t \in[0, T]\). Then, we can proceed the proof as in [5], Proposition 2.1. By regarding f in [5], Proposition 2.1, as \([y, v] + f\) in Eq. (1.1), we can infer by [5], Proposition 2.1, that the weak solution y of Eq. (1.1) satisfies

By Lemma 2.1, (2.4), and (3.2) we can obtain for every fixed \(t \in[0, T]\) that

where \({\hat{y}}'(\cdot) = y'(\cdot) {\mathcal {X}}_{[0, t]} (\cdot)\).

Thus, we have (3.4). □

It is verified from the assumptions on f and k that the right-hand side of (3.4) is continuous in t. Hence, we have that \(t \to \Vert \nabla y'(t) \Vert + \Vert \Delta y(t) \Vert \) is continuous on \([0, T]\). Therefore, as in the proof of Lions and Magenes [15], p.279, we have

Theorem 3.1

Assume that \((y_{0}, y_{1} ) \in H^{2}_{0} \times H^{1}_{0}\), \(k \in C^{1}([0, T])\), and \(f \in L^{2}(0, T; L^{2})\). Then Eq. (1.1) has a unique weak solution y in \({ S}(0,T) \equiv W (0, T) \cap C([0, T]; H^{2}_{0}) \cap C^{1} ([0, T]; H^{1}_{0})\). Moreover, the solution mapping \(p=(y_{0}, y_{1}, f) \to( y(p), y_{t} (p), v(p))\) of \({\mathcal {P}} \equiv H^{2}_{0} \times H^{1}_{0} \times L^{2}(0, T; L^{2}) \) into \(C([0, T]; H^{2}_{0} ) \times C([0, T]; H^{1}_{0}) \times C([0, T]; W^{2, \infty}) \) is locally Lipschitz continuous.

Indeed, let \(p_{1}= (y_{0}^{1}, y^{1}_{1}, f_{1}) \in{\mathcal {P}}\) and \(p_{2}= (y_{0}^{2}, y^{2}_{1}, f_{2}) \in{\mathcal {P}}\). We prove Theorem 3.1 by showing the inequality

where \(C > 0\) is a constant depending on data, and

We will omit writing the integral variables in the definite integral without any confusion.

Proof of Theorem 3.1

For the well-posedness of weak solutions of Eq. (1.1), we can refer to [3, 4] (without memory term in Eq. (1.1)) and [2] (with memory term but without viscosity damping term \(- \Delta y_{tt}\) in Eq. (1.1)). As explained in [2], von Kármán nonlinearity is subcritical; thus, the issues of well-posedness and regularity of weak solutions are standard. Therefore, combining those results in [3, 4] and [2], we can deduce that Eq. (1.1) possesses a unique weak solution \(y \in{ S}(0,T) \) under the data condition \(p=(y_{0}, y_{1}, f) \in H^{2}_{0} \times H^{1}_{0} \times L^{2}(0, T; L^{2}) \) such that

Based on this result, we prove inequality (3.7). For this purpose, we denote \(y_{1} - y_{2} \equiv y(p_{1})- y(p_{2})\) by ψ and \(v_{1} - v_{2} \equiv v(p_{1})- v(p_{2})\) by V. Then, we can get from Eq. (1.1) that ψ and V satisfy the following equation in weak sense:

We note that

In view of Eq. (3.5), corresponding to Eq. (1.1), we can get that the weak solution ψ of Eq. (3.9) satisfies

The right-hand side of (3.11) can be estimated as follows:

By Lemma 2.2 we can obtain the following:

We replace the right-hand side of (3.11) by the right members of (3.12)-(3.17) to obtain

By applying Poincaré’s and Gronwall’s inequality to (3.18) we have

Also, for almost \(t \in[0, T]\), we have

By (3.19) and (3.20) we can deduce

Finally, by combining (3.19) and (3.21) we obtain (3.7).

This completes the proof. □

Let \({\mathcal {U}}\) be a Hilbert space of control variables, and let B be an operator,

called a controller.

We consider the following nonlinear control system:

where \(y_{0} \in H^{2}_{0}\), \(y_{1} \in H^{1}_{0}\), and \(u \in{ \mathcal {U}}\) is a control. By Theorem 3.1 and (4.1) we can define uniquely the solution map \(u\to y(u)\) of \({ \mathcal {U}}\) into \(S(0, T)\). The observation of the state is assumed to be given by

where C is an operator called the observer, and M is a Hilbert space of observation variables. The quadratic cost function associated with the control system (4.2) is given by

where \(Y_{d} \in M\) is a desired value of \(y(u)\), and \(R \in{ \mathcal {L}}({ \mathcal {U}},{\mathcal {U}}) \) is symmetric and positive, that is,

for some \(d > 0\). Let \({\mathcal {U}}_{\mathrm{ad}}\) be a closed convex subset of \({\mathcal {U}}\), which is called the admissible set. An element \(u^{*} \in {\mathcal {U}}_{\mathrm{ad}}\) that attains the minimum of J over \({\mathcal {U}}_{\mathrm{ad}}\) is called an optimal control for the cost (4.4).

Existence of an optimal control

As indicated in Introduction, we need to show the existence of an optimal control and to give its characterization. The existence of an optimal control \(u^{*}\) for the cost (4.4) can be stated by the following theorem.

Theorem 4.1

Assume that the hypotheses of Theorem  3.1 are satisfied. Then there exists at least one optimal control u for the control problem (4.2) with the cost (4.4).

Proof

Set \(J_{0}= \inf_{u \in{ \mathcal {U}}_{\mathrm{ad}}}J(u)\). Since \({\mathcal {U}}_{\mathrm{ad}}\) is nonempty, there is a sequence \(\{ u_{n} \}\) in \({\mathcal {U}}\) such that

Obviously, \(\{J(u_{n})\}\) is bounded in \({\mathbf {R}}^{+}\). Then by (4.5) there exists a constant \(K_{0}>0\) such that

This shows that \(\{u_{n}\}\) is bounded in \({\mathcal {U}}\). Since \({\mathcal {U}}_{\mathrm{ad}}\) is closed and convex, we can choose a subsequence (denoted again by \(\{ u_{n} \}\)) of \(\{u_{n}\}\) and find \(u \in{\mathcal {U}}_{\mathrm{ad}} \) such that

as \(n \rightarrow\infty\). From now on, each state \(y_{n}=y(u_{n}) \in{ S}(0,T)\) corresponding to \(u_{n}\) is a solution of

By (4.6) the term \(Bu_{n}\) is estimated as

Hence, noting that \(y(0,0,0,t) =0\) and \(v(0,0,0,t)=0\), it follows from Theorem 3.1 that

By (4.10) we easily verify that \([y_{n}, v_{n}]\) is bounded in \(L^{2}(0, T;L^{2})\). Therefore, by the extraction theorem of Rellich we can find a subsequence of \(\{ y_{n}\}\), say again \(\{ y_{n} \}\), and find \(y\in { W}(0,T) \cap L^{\infty}(0, T; H^{2}_{0}) \) with \(y' \in L^{\infty} (0, T; H^{1}_{0})\) and \(F \in L^{2}(0, T; L^{2})\) such that

To prove \(F = [y, -G[y,y]]\), we employ the idea given in Dautray and Lions [7]. By similar manipulations given in Dautray and Lions [7], pp.564-566, we can deduce that the weak limit y in (4.11) is a weak solution of the linear problem

As in (3.5), the weak solution y of Eq. (4.15) satisfies the following energy equality:

We can also deduce, as in (3.5), that the weak solution \(y_{n}\) of Eq. (4.8) satisfies the following energy equality:

We note the following simple equalities:

Adding (4.16) to (4.17), denoting \(y_{n} - y\) by \(\phi_{n}\), and using the above equalities, we have

where

Then by routine calculations in (4.18), as in the proof of Theorem 3.1, we derive the inequality

By virtue of (4.11)-(4.13) together with [7], pp.518-520, we can extract a subsequence \(\{ y_{n_{k} } \}\) of \(\{ y_{n} \}\) such that, as \(k \to \infty\),

Since the imbedding \(H^{1}_{0} \hookrightarrow L^{2}\) is compact, by virtue of (4.11), we can refer to the result of the Aubin-Lions-Temam compact imbedding theorem (see Temam [16]; p.271) to verify that \(\{ y'_{n} \}\) is precompact in \(L^{2}(0,T; L^{2})\). Hence, there also exists a subsequence \(\{y'_{n_{k}} \} \subset\{ y'_{n} \}\) such that

From (4.7), (4.14), and (4.30) we have

In view of (4.16), the sum of (4.19) and all the limits from (4.26) to (4.29) and (4.31) are 0, so that

Therefore, from (4.25) and (4.32) we get that

Thus, by Lemma 2.2, Theorem 3.1, and (4.33) it follows that

as \(k \to \infty\), where \(p^{*} = (y_{0}, y_{1}, Bu^{*})\) and \(p_{n_{k}} = (y_{0}, y_{1}, Bu_{n_{k}})\). Hence, by the uniqueness of the weak limits, from (4.14) and (4.34) it follows that

We replace \(y_{n}\) by \(y_{n_{k}}\) and take \(k\to\infty\) in (4.8). Then by the standard argument in Dautray and Lions ([7], pp.561-565) we conclude that the limit y is a weak solution of

Also, since Eq. (4.36) has a unique weak solution \(y \in{S}(0,T)\) by Theorem 3.1, we conclude that \(y=y(u^{*})\) in \({ S}(0,T)\) by the uniqueness of solutions, which implies that \(y(u_{n})\to y(u^{*})\) weakly in \({ W}(0,T)\). Since C is continuous on \(S(0, T) \subset{ W}(0,T)\) and \(\Vert \cdot \Vert _{M}\) is lower semicontinuous, it follows that

It is also clear from \(\liminf_{k\rightarrow\infty} \Vert R^{\frac{1}{2}}u_{n}\Vert _{ \mathcal {U}} \geq \Vert R^{\frac{1}{2}}u^{*} \Vert _{\mathcal {U}}\) that \(\liminf_{k\rightarrow\infty} (Ru_{n},u_{n})_{\mathcal {U}}\geq (Ru^{*},u^{*})_{\mathcal {U}}\). Hence,

But since \(J(u^{*})\geq J_{0}\) by definition, we conclude that \(J(u^{*})=\inf_{u\in{\mathcal {U}}_{\mathrm{ad}}}J(u)\). This completes the proof. □

In this section, we shall characterize the optimal controls by giving necessary conditions for optimality. For this, it is necessary to write down the necessary optimality condition

and to analyze (4.37) in view of the proper adjoint state system, where \(DJ(u^{*})\) denotes the Gâteaux derivative of \(J(u)\) at \(u=u^{*}\). That is, we have to prove that the mapping \(u \to y(u)\) of \({\mathcal {U}} \to{ S}(0,T)\) is Gâteaux differentiable at \(u=u^{*}\). First, we can see the continuity of the mapping.

Lemma 4.1

Let \(w\in{\mathcal {U}}\) be arbitrarily fixed. Then

Proof

The proof is an immediate consequence of Theorem 3.1. □

The solution map \(u \to y(u)\) of \({\mathcal {U}}\) into \({ S}(0,T)\) is said to be Gâteaux differentiable at \(u=u^{*}\) if for any \(w\in {\mathcal {U}}\), there exists a \(Dy(u^{*})\in{\mathcal {L}}({\mathcal {U}}, {S}(0,T))\) such that

The operator \(Dy(u^{*})\) denotes the Gâteaux derivative of \(y(u)\) at \(u=u^{*}\), and the function \(Dy(u^{*})w \in{ S}(0,T)\) is called the Gâteaux derivative in the direction \(w\in{\mathcal {U}}\), which plays an important part in the nonlinear optimal control problem.

Theorem 4.2

The map \(u\to y(u)\) of \({\mathcal {U}}\) into \({ S}(0,T)\) is Gâteaux differentiable at \(u=u^{*}\) and such the Gâteaux derivative of \(y(u)\) at \(u=u^{*}\) in the direction \(u-u^{*}\in{\mathcal {U}}\), say \(z=Dy(u^{*})(u-u^{*})\), is a unique weak solution of the following problem:

Proof

Let \(\lambda\in(-1,1)\), \(\lambda\ne0\). We set \(y_{\lambda}:= y(u^{*} +\lambda(u-u^{*}))\) and

Then, in the weak sense, \(z_{\lambda}\) satisfies

where

Here we note that

Thus, from (2.5), Theorem 3.1, and (4.41) we deduce

where \(p_{\lambda} = (y_{0}, y_{1}, B(u^{*} + \lambda(u-u^{*})) ) \) and \(p^{*} = (y_{0}, y_{1}, B u^{*} )\). Hence, by considering the energy equality satisfied by \(z_{\lambda}\) like (3.5) we get from (4.42) and the proof of Theorem 3.1 that the weak solution \(z_{\lambda}\) of Eq. (4.40) satisfies

Therefore, from (4.42) and (4.43) we see that there exists \(z \in{ W}(0,T) \cap L^{\infty} (0, T; H^{2}_{0})\) with \(z' \in L^{\infty}(0, T; H^{1}_{0})\), \(F \in L^{2} (0, T; L^{2})\) and a sequence \(\{\lambda_{k}\} \subset(-1,1)\) tending to 0 such that, as \(k \to \infty\),

We replace \(z_{\lambda}\) by \(z_{\lambda_{k}}\) and take \(k\to\infty\) in Eq. (4.40). Then by the standard argument in Dautray and Lions ([7], pp.561-565) we conclude that the limit z is a weak solution of

Using (4.44)-(4.47), the respective energy equalities of Eq. (4.40) with \(z_{\lambda}\) replaced by \(z_{\lambda_{k}}\), and Eq. (4.48), we can proceed as in the proof of Theorem 4.1 to obtain

By Theorem 3.1 and Lemma 2.2 we can verify the following:

where \(p_{\lambda_{k}} = (y_{0}, y_{1}, B(u^{*} + \lambda_{k} (u-u^{*})) ) \) and \(p^{*} = (y_{0}, y_{1}, B u^{*} )\). Thus, from Lemma 4.1, (4.49), and (4.50), we have

as \(k \to\infty\).

Similarly, we can also show that

as \(k \to\infty\). Therefore, by (4.51) and (4.52) we can show that

as \(k \to\infty\).

Consequently, we can infer from (4.47) and (4.52) that

Hence, it readily follows from (4.49) and (4.54) that \(z_{\lambda_{k}}\to z=Dy(u^{*})(u-u^{*})\) strongly in \({S}(0,T)\) as \(k \rightarrow\infty\), in which z is a weak solution of (4.39).

This completes the proof. □

Theorem 4.2 means that the cost \(J(u)\) is Gâteaux differentiable at \(u^{*}\) in the direction \(u-u^{*}\) and the optimality condition (4.37) is rewritten by

where \(\Lambda_{M}\) is the canonical isomorphism M onto \(M'\).

In this paper, we consider the following physically important observation. We take \(M=L^{2}(0, T; L^{2})\) and \(C \in{\mathcal {L}}({ W}(0,T), M)\) and observe that \(Cy(u)= y(u; \cdot) \in L^{2}(0, T; L^{2})\).

Necessary condition of an optimal control for distributive observation

In this subsection, we consider the cost functional expressed by

where \(Y_{d} \in L^{2}(0, T;L^{2})\) is the desired value. Let \(u^{*}\) be the optimal control subject to (4.2) and (4.56). Then the optimality condition (4.55) is represented by

where z is the weak solution of Eq. (4.39). Now we formulate the adjoint system to describe the optimality condition:

Proposition 4.1

Equation (4.58) admits a unique solution \(p(u^{*}) \in S(0, T)\).

Proof

Since

the time reversed equation of Eq. (4.58) (\(t \to T-t\) in Eq. (4.58)) is given by

where \(\psi(t)= p(u^{*};T-t)\).

Here we note that, like (4.42),

where \(p^{*} = (y_{0}, y_{1}, Bu^{*})\). Thus, by Theorem 3.1 and [5], the conditions \(Y_{d} \in L^{2}(0, T;L^{2})\) and (4.60) enable us to deduce that there exists a unique \(\psi\in S(0, T)\).

This completes the proof. □

Now we proceed the calculations. We multiply both sides of the weak form of Eq. (4.58) by z and integrate it over \([0,T]\). Then we have

By Fubini’s theorem we have

By Lemma 2.1 we deduce

We observe that by considering \(\phi, \psi\in H^{2}_{0}\) we have \([\phi, \psi] \in L^{1}\). However, since \(n=2\), we have

and, therefore,

Thus, since G is a self-adjoint operator, by Lemma 2.1 and (4.65) we have

Considering from (4.62) to (4.66), the terminal value conditions of p in (4.58), and Eq. (4.39), we can verify by integration by parts that the left-hand side of (4.61) yields

Therefore, combining (4.61) and (4.67), we deduce that the optimality condition (4.57) is equivalent to

Hence, we give the following theorem.

Theorem 4.3

The optimal control \(u^{*}\) for (4.56) is characterized by the following system of equations and inequality:

Local uniqueness of an optimal control

We note that the uniqueness of an optimal control in nonlinear equation is not ensured. However, it is worth noticing partial results. For instance, we can refer to the result in [12] to obtain the local uniqueness of an optimal control for distributive observation case. For that reason, in this subsection, we take \(M = L^{2}((0, t) \times\Omega)\) and observe that \(y \in L^{2}((0, t) \times\Omega)\). Hence, we consider the following quadratic cost functional:

where \(Y_{d} \in L^{2}((0, t) \times\Omega)\).

In order to show the local uniqueness of an optimal control by making use of the strict convexity of quadratic cost (see [17]), we consider the following proposition.

Proposition 4.2

The map \(w\to y(w)\) of U into \({S}(0,T)\) is second-order Gâteaux differentiable at \(w=u\) and such the second-order Gâteaux derivative of \(y(w)\) at \(w=u\) in the direction \(w-u\in{\mathcal {U}}\), say \(g =D^{2} y(u)(w-u, w-u)\), is a unique solution of the following problem:

where

and z is the weak solution of Eq. (4.39), changing \(B(u-u^{*})\) by \(B(w-u)\).

Proof

The proof is similar to that of Theorem 4.2. □

Lemma 4.2

Let g be the weak solution of Eq. (4.69). Then we can show that

where \(C > 0 \) is a constant depending on the time T and the data conditions of the equation of \(y(u)\).

Proof

Let z be the solution of Eq. (4.39), changing with \(B(u-u^{*})\) to \(B(w-u)\). Then, using the same arguments as in Eq. (3.1), we can deduce that

Also, for the solution g of Eq. (4.69), we can show that

where \(p = (y_{0}, y_{1}, Bu)\). Combining (4.71) with (4.72), we have (4.70). □

We prove the local uniqueness of the optimal control.

Theorem 4.4

When t is small enough, there is a unique optimal control for the cost (4.68).

Proof

We show the local uniqueness by proving the strict convexity of the map \(u \in{\mathcal {U}}_{\mathrm{ad}} \to J(u)\). Therefore, as in [17], we need to show, for all \(u, w \in{\mathcal {U}}_{\mathrm{ad}}\) (\(u \ne w\)),

For simplicity, we denote \(y(u + \xi(w-u))\), \(z(u + \xi(w-u))\), and \(g(u + \xi(w-u))\) by \(y(\xi)\), \(z(\xi)\), and \(g(\xi)\), respectively. We calculate

From (4.74) we obtain the second Gâteaux derivative of J:

By Lemma 4.2 and (4.75) we deduce that

Here we can take \(t > 0 \) small enough so that the right-hand side of (4.76) is strictly greater than 0. Therefore, we obtain the strict convexity of the quadratic cost \(J(u)\), \(u \in{\mathcal {U}}_{\mathrm{ad}}\), which proves this theorem. □

Remark 4.1

If we assume that d is large enough, then we can obtain the strict convexity of the quadratic cost (4.68) in the global sense. Therefore, we can obtain the desired result of Theorem 4.4 in the global sense for the cost (4.68).

This research was supported by the Daegu University Research Grant 2013.

Affiliations

1. Department of Mathematics Education, College of Education, Daegu University, Jillyang, Gyeongsan, Gyeongbuk, Republic of Korea

   • Jinsoo Hwang

Corresponding author

Correspondence to Jinsoo Hwang.

Competing interests

The author declares to have no completing interests.

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