We consider a damped Kirchhoff-type equation with Dirichlet boundary conditions. The objective is to show the Fréchet differentiability of a nonlinear solution map from a bilinear control input to the solution of a Kirchhoff-type equation. We use this result to formulate the minimax optimal control problem. We show the existence of optimal pairs and find their necessary optimality conditions.

1. Introduction

Let Ω be an open bounded set of Rⁿ(n ≤ 3) with a smooth boundary Γ. We set Q = (0, T) × Ω, Σ = (0, T) × Γ for T > 0. We consider a strongly damped Kirchhoff-type equation described by the following Dirichlet boundary value problem:

(1)

where ^′ = ∂/∂t, y is the displacement of a string (or membrane), μ > 0, f is a forcing function, and U is a bilinear forcing term, which is usually a bilinear control variable that acts as a multiplier of the displacement term. |·| denotes the Euclidean norm on Rⁿ. As is well known by Kirchhoff [1], the nonlinear part of (1) represents an extension effect of a vibrating string (or membrane). Many kinds of Kirchhoff-type equations have been research subject of many researchers (see Arosio [2], Spagnolo [3], Pohozaev [4], Lions [5], Nishihara and Yamada [6], and references therein).

From a physical perspective, the damping of (1) represents an internal friction in an elastic string (or membrane) that makes the vibration smooth. Therefore, we can obtain the well-posedness in the Hadamard sense under sufficiently smooth initial conditions (see [7]). Based on this result, Hwang and Nakagiri [8] set up optimal control problems developed by Lions [9] with (1) using distributed forcing controls. They proved the Gâteaux differentiability of the quasilinear solution map from the control variable to the solution and applied the result to derive the necessary optimality conditions for optimal control in some observation cases.

It is important and challenging to extend the optimal control theory to practical nonlinear partial differential equations. There are several studies on semilinear partial differential equations (see [10]). Indeed, the extension of the theory to quasilinear equations is much more restrictive because the differentiability of a solution map is quite dependent on the model due to the strong nonlinearity. Only a few studies have investigated this topic (see [8, 11, 12]). Thus, the differentiability of a solution map in any sense is important to study optimal control or identification problems. In most cases, Gâteaux differentiability may be enough to solve a quadratic cost optimal control problem as in [8]. However, to study the problem in more general cost function like nonquadratic or nonconvex functions, the Fréchet differentiability of a solution map is more desirable.

In this paper, we show the Fréchet differentiability of the solution map of (1): U → y from the bilinear control input variables to the solutions of (1). In the author’s knowledge, the Fréchet differentiability of a quasilinear solution map is not studied yet. Based on the result, we construct and solve a bilinear minimax optimal control problem on (1). For the study, we refer to the linear results from Belmiloudi [13], in which the author considered some linear parabolic partial differential equations as the state equations for the problem. Minimax control framework has been used by many researchers for various control problems. There are many literatures related to the minimax control problems. We can refer to just a few: Arada and Raymond [14], Lasiecka and Triggiani [15], and Li and Yong [16].

In this paper, the minimax control framework was employed to take into account the undesirable effects of system disturbance (or noise) in control inputs such that a cost function achieves its minimum even when the worst disturbances of the system occur. For this purpose, we replace the bilinear multiplier U in (1) by u + v, where u is a control variable that belongs to the admissible control set

and v is a disturbance (or noise) that belongs to the admissible disturbance set

. We introduce the following cost function to be minimized within

and maximized within

(2)

where y is a solution of (1), M is a Hilbert space of observation variables,

is an operator from the solution space of (1) to M, Y_d ∈ M is a desired value, and the positive constants α and β are the relative weights of the second and third terms on the RHS of (2).

As mentioned, another goal of this paper is to find and characterize the optimal controls of the cost function (2) for the worst disturbances through control input in (1). This leads to the problem of finding the saddle points of the cost function (2). First, we prove the existence of an admissible control

and disturbance (or noise)

such that (u^∗, v^∗) is a saddle point of the functional J(u, v) of (2). That is,

(3)

Secondly, we derive an optimality condition for (u^∗, v^∗) in (3). In this paper, we use the terminology optimal pair to represent such a saddle point (u^∗, v^∗) in (3). To prove the existence of an optimal pair (u^∗, v^∗) satisfying (3), we follow the arguments given by Belmiloudi [13], in which the author employed the minimax theorem in infinite dimensions given by Barbu and Precupanu [17]. Next, we derive the necessary optimal conditions for some observation cases that should be satisfied by the optimal pairs in these observation cases. To derive these conditions, we refer to the studies about bilinear optimal control problems where the state equation is linear partial differential equations such as the reaction diffusion equation or Kirchhoff plate equation (see [13, 18–20] and references therein).

We now explain the content of this paper. In Section 2, we prove the well-posedness of (1) in the Hadamard sense under sufficiently smooth initial conditions, including a stability estimate from the data space to the solution space. In Section 3, we shall show that the solution map of (1): U → y is Fréchet differentiable. In Section 4, we shall study the minimax optimal control problems: By using the Fréchet differentiability of the solution maps u → y and v → y, we prove that the maps u → J and v → J are convex and concave, respectively, under the assumptions that α, β are sufficiently large. And with an assumption on the operator in (2), we prove the maps u → J and v → J are lower and upper semicontinuous, respectively. As a result, we can prove the existence of an optimal pair. Next, we derive the necessary optimal conditions for some practical observation cases by employing associate adjoint systems. Especially, we use a first-order Volterra integrodifferential equation as a proper adjoint equation in the velocity’s observation case, which is another novelty of this paper.

2. Preliminaries

Throughout this paper, we use C as a generic constant. Let X be a Banach space. We denote its topological dual as X^′ and the duality pairing between X^′ and X by

. We also introduce the following abbreviations:

(4)

where p ≥ 1.

is the completions of

in H^k for k ≥ 1. Let the scalar product on L² be (·,·)₂. From Poincare’s inequality and the regularity theory for elliptic boundary value problems (cf. Temam [21, p. 150]), the scalar products on

and

can be endowed as follows:

(5)

(6)

Then we know that

(7)

The duality pairing between

and H⁻¹ is denoted by 〈ϕ, ψ〉 _1,−1. It is clear that

(8)

Each space is dense in the following one, and the injections are continuous and compact. According to Adams [22], we know that the embeddings

(9)

(10)

are compact when n ≤ 3.

The solution space S(0, T) of (1) of strong solutions is defined by

(11)

which is endowed with the norm

(12)

where g^′ and g^′′ denote the first and second order distributional derivatives of g.

Definition 1. A function y is said to be a strong solution of (1) if y ∈ S(0, T) and y satisfies

(13)

From Dautray and Lions [23, p.480] and Lions and Magnes [24], we remark that

(14)

The following variational formulation is used to define the weak solution of (1).

Definition 2. A function y is said to be a weak solution of (1) if and y satisfies

(15)

The following is the well-known Gronwall inequality.

Lemma 3. Let η(·) be a nonnegative, absolutely continuous function on [0, T], which satisfies the following differentiable inequality for a.e. t ∈ [0, T]:

(16)

where ϕ and ψ are nonnegative, summable functions on [0, T]. Then

(17)

Proof. See Evans [25, p.624].

Throughout this paper, we will omit writing the integral variables in the definite integral without any confusion. Referring to [7] and the previous result of [8], we can obtain the following theorem on existence, uniqueness, and regularity of a solution of (1).

Theorem 4. Assume that , and U ∈ L^∞(Q). Then (1) has a unique strong solution y ∈ S(0, T). Moreover, the solution mapping p = (y₀, y₁, f, U) → y(p) of into S(0, T) is locally Lipschitz continuous. Let and The following is satisfied:

(18)

where C > 0 is a constant depending on the data.

Proof. From [7], for each fixed U ∈ L^∞(Q) in (1), we can infer that (1) admits a unique strong solution y ∈ S(0, T) under the data condition

Based on this result, for each and , we prove the inequality (18). For that purpose, we denote y₁ − y₂ ≡ y(p₁) − y(p₂) by ψ. Then, from (1), we can know that ψ satisfies the following:

(19)

where

(20)

In estimating ψ in (19), we can refer to the previous results [8, Theorem 2.1] to obtain the following inequality:

(21)

Since

and S(0, T)↪L²(Q), we have

(22)

Together with (21) and (22), we can deduce the following:

(23)

Applying (23) to (19), we have

(24)

From (23) and (24), we can obtain

(25)

This completes the proof.

Corollary 5. For , the following inequality is satisfied:

(26)

where C > 0 is a constant depending on the data and y(p₁) and y(p₂) are the solutions of (1) corresponding to p₁ and p₂, respectively.

Proof. We denote y(p₁) − y(p₂) by ψ. Then, as in the proof of Theorem 4, we can know that ψ satisfies the following:

(27)

where ϵ(ψ) is given in (20). Estimating ψ in (27) as in the proof of Theorem 4, we can arrive at

(28)

Thanks to the fact that y(p₂) ∈ S(0, T)↪C([0, T]; D(Δ)) and (10), we can know that

. Thus we have

(29)

Consequently, from (28) and (29), we have (26).

This completes the proof.

3. Fréchet Differentiability of the Nonlinear Solution Map

In this section, we study the Fréchet differentiability of the nonlinear solution map. The Fréchet differentiability of the solution map plays an important role in many applications. Let

We consider the nonlinear solution map from

to y(u) ∈ S(0, T), where y(u) is the solution of

(30)

Based on Theorem 4, for fixed

, we know that the solution map

, which maps from the term

of (30) to y(u) ∈ S(0, T), is well defined and continuous. We define the Fréchet differentiability of the nonlinear solution map as follows.

Definition 6. The solution map u → y(u) of into S(0, T) is said to be Fréchet differentiable on if for any there exists a such that, for any ,

(31)

The operator T(u) is called the Fréchet derivative of y at u, which we denote by Dy(u), and T(u)w = Dy(u)w ∈ S(0, T) is called the Fréchet derivative of y at u in the direction of

Theorem 7. The solution map u → y(u) of to S(0, T) is Fréchet differentiable on and the Fréchet derivative of y(u) at u in the direction , that is to say z = Dy(u)w, is the solution of

(32)

We prove this theorem by two steps:

(i)
For any , (32) admits a unique solution z ∈ S(0, T). That is, there exists an operator satisfying Tw = z( = z(w)).
(ii)
We show that as

Proof. (i) Let

(33)

Then from Theorem 4 and (14), we can estimate the above as follows:

(34)

Hence, by (34) we know that

(35)

To estimate the solution z of (32), we take the scalar product of (32) with −Δz^′ − Δz in L²:

(36)

Integrating (36) over [0, t], we obtain

(37)

The right hand side of (37) can be estimated as follows:

(38)

(39)

(40)

(41)

Considering (38)-(41) and taking ϵ = (1/6)min⁡{1/2, μ/2}, we can obtain the following from (37):

(42)

Applying Lemma 3 to (42), we obtain

(43)

In view of (32), (43) implies that

(44)

Therefore, from (43) and (44), we can know that z ∈ S(0, T), and the solution z( = z(w)) of (32) satisfies

(45)

Hence, from (45), the mapping

is linear and bounded. From this, we can infer that there exists

such that Tw = z(w) for each

(ii) We set the difference δ = y(u + w) − y(u) − z. Then, from (30) and (32), we can have the following:

(46)

Thus, we know from (46) that δ satisfies

(47)

where

(48)

If we let

(49)

then by similar arguments used for (34), we have

(50)

Thanks to (50), if we follow similar arguments as in (i), then we can arrive at

(51)

From (14), Theorem 4, and (45), we can deduce the following:

(52)

(53)

(54)

Hence, from (51) to (54), we can obtain

(55)

which immediately implies that

This completes the proof.

The following result plays an important role in proving the existence of optimal controls in the next section.

Proposition 8. Given , the Fréchet derivative Dy(u)w is locally Lipschitz continuous on with L²(Q) topology. Indeed, it is satisfied that

(56)

where C > 0 is a constant depending on the data.

Proof. Let z_i = Dy(u_i)w, (i = 1,2) be the solutions of (32) corresponding to u_i, (i = 1,2), and we set ϕ = z₁ − z₂. Then, by similar calculations as in (46), we can deduce that ϕ satisfies

(57)

where

(58)

By similar arguments as in the proof of (i) of Theorem 7, ϕ in (57) can be estimated as follows:

(59)

From Theorem 4, the embedding

, and the first inequality of (45), we can deduce

(60)

We can estimate I_i (i = 1, …, 4) of (57) as follows:

(61)

(62)

(63)

(64)

From (61) to (64), we can obtain the following from (59):

(65)

This completes the proof.

4. Quadratic Cost Minimax Control Problems

In this section, we study the quadratic cost minimax optimal control problems for a damped Kirchhoff-type equation. Let the following be the set of the admissible controls:

(66)

Let the following be the set of the admissible disturbance or noises:

(67)

To perform our variational analysis, L²(Q) norms of

and

are preferable, even though

and

are subsets of

For simplicity, let

be a product space defined by

Using Theorem 4, we can uniquely define the solution mapping

, which maps the term

to the solution y(q) ∈ S(0, T), which satisfies the following equation:

(68)

The solution y(q) of (68) is the state of the control system (68). From Theorem 7, we can deduce that the map q = (u, v) → y(q) of

to S(0, T) is Fréchet differentiable at q = q^∗ = (u^∗, v^∗), and the Fréchet derivative of y(q) at q = q^∗ in the direction

, say z = Dy(q^∗)w is a unique solution of the following problem:

(69)

The quadratic cost function associated with the control system (68) is

(70)

where M is a Hilbert space of observation variables, the operator

is an observer, Y_d ∈ M is a desired value, and the positive constants α and β are the relative weights of the second and the third terms on the RHS of (70).

To pursue our objective, we assume that the observer

in (70) is a compact operator. As mentioned in the introduction, the minimax optimal control problem can be summarized as follows:

(i)
Find an admissible control and a noise (or disturbance) such that (u^∗, v^∗) is a saddle point of the functional J(u, v) of (70). That is,
(71)
(ii)
Characterize (u^∗, v^∗) (optimality condition).

Such a pair (u^∗, v^∗) in (71) is called an optimal pair (or an optimal strategy pair) for the problem (70).

4.1. Existence of Optimal Pairs

To study the existence of optimal pairs, we present the following results.

Proposition 9. The solution mapping from to S(0, T) is continuous from the weakly-star topology of to the weak topology of S(0, T).

In proving the Proposition 9, we need the following compactness lemma.

Lemma 10. Let X, Y and Z be Banach spaces such that the embeddings X↪Y↪Z are continuous and the imbedding X↪Y is compact. Then a bounded set of W^1,∞(0, T; X, Z) = {g∣g ∈ L^∞(0, T; X), g^′ ∈ L^∞(0, T; Z)} is relatively compact in C([0, T]; Y).

Proof. See Simon [26].

Proof of Proposition 9. Let and let be a sequence such that

(72)

For simplicity, we let each state y_n = y(q_n) be a solution of

(73)

We conduct the scalar product of (73) with

in L²:

(74)

which immediately implies

(75)

The integration of (75) over [0, t] implies

(76)

where

(77)

By conducting similar calculations to the proof of (i) of Theorem 7, we can obtain the following from (76):

(78)

Since we know from Theorem 4 that y_n ∈ S(0, T), we can note that

(79)

From (78) and (79), we can infer

(80)

Applying Lemma 3 to (80), we have

(81)

Theorem 4 and (81) imply that y_n remains in a bounded set of

Therefore, by using Rellich’s extraction theorem, we can find a subsequence of {y_n} also called {y_n}, and find

such that

(82)

(83)

(84)

Since the embedding

is compact, we can apply Lemma 10 to (83) and (84) with X = D(Δ) and

in Lemma 10 to verify that

(85)

Hence, we can find a subsequence

if necessary such that

(86)

Therefore, (82) and (86) imply

(87)

From (72) and (85), we can also extract a subsequence, if necessary, denoted again by q_n ≡ (u_n, v_n) such that

(88)

We replace y_n by

, if necessary, and take k → ∞ in (73). Then, by the standard argument in Dautray and Lions [23, pp.561-565], we conclude that the limit y is a solution of

(89)

Moreover, from the uniqueness of solutions of (89), we conclude that y = y(q) in S(0, T), which implies that y(q_n)⇀y(q) weakly in S(0, T).

This completes the proof.

We now study the existence of optimal pairs.

Theorem 11. Let the observer in (70) be a compact operator. Then, for sufficiently large α and β in (70), there exists such that (u^∗, v^∗) satisfies (71).

Proof. Let be the map u → J(u, v) and let be the map v → J(u, v). To obtain the existence of optimal pairs in the minimax control problem, we follow the steps given by [13]: We prove that is convex and lower semicontinuous for all and that is concave and upper semicontinuous for all Then, we employ the minimax theorem in infinite dimensions (see Barbu and Precupanu [17]).

For sufficiently large α and β in (70), we first prove the convexity of and the concavity of . To prove the convexity of , which is a differentiable map, it is sufficient to show that

(90)

From Fréchet differentiability of the solution map u → y(u, v), where v is fixed, (90) can be rewritten as

(91)

where D_uy(u_i, v)(u₁ − u₂), (i = 1,2) are solutions of (69), in which (u^∗ + v^∗)z + (h + l)y(p) is replaced by (u_i + v)z + (u₁ − u₂)y(u_i, v), (i = 1,2), respectively. We can easily deduce that (91) is equivalent again to

(92)

From Corollary 5, Proposition 8, and (60), we can estimate the left hand side of (92) as follows:

(93)

(94)

Considering from (92) to (94), we can deduce that there exists a sufficiently large

such that, for any

, (92) holds true. Therefore, the map

is convex.

Similarly, we can also show that there exist a sufficiently large such that the following inequality is satisfied for any :

(95)

This also indicates the concavity of

Next, we prove the existence of an optimal pair by verifying that is lower semicontinuous for all and is upper semicontinuous for all Let be a minimizing sequence of J. Thus

(96)

Since

defined by (66) is a closed, bounded, and convex in

, we can extract a subsequence

such that

(97)

Then, by Proposition 9, we have

(98)

Thus, by the assumption that

is a compact operator, we can extract a subsequence of

, if necessary, denoted again by

, such that

(99)

From (97), it can be easily verified for the same subsequence

in (97) that

(100)

Due to the weakly lower semicontinuity in the L²(Q) norm topology, we can determine from (99) and (100) that the map

is lower semicontinuous for all

By similar arguments, we can prove that

is upper semicontinuous for all

Hence, we know that

(101)

But since J₀(v) ≤ J(u^∗, v), we have

(102)

Similarly, we also know that there exists

such that

(103)

From (102) and (103), we can conclude that

is an optimal pair for the cost (70).

This completes the proof.

4.2. Necessary Conditions of Optimal Pairs

We now turn to the necessary optimality conditions that have to be satisfied by optimal pairs with the cost (70). For this purpose, we consider the following two types of observations C_i, (i = 1,2) of distributive and terminal values:

(1)
we take M₁ = L²(Q) × L² and and observe C₁y(q) = (y(q; ·), y(q; T)) ∈ L²(Q) × L²;
(2)
we take M₂ = L²(Q) and and observe C₂y(q) = y^′(q; ·) ∈ L²(Q).

Remark 12. Clearly, the embedding S(0, T)↪L²(Q) is compact. From the embedding (14) we can utilize Lemma 10 in which X = D(Δ) and Y = Z = L² to obtain the embedding S(0, T)↪C([0, T]; L²) is also compact. Consequently, the observer C₁ is a compact operator. Thus, C₁ satisfies the requirement for the existence of optimal pairs given in Theorem 11.

Remark 13. Since y^′(q) ∈ H¹(0, T; D(Δ), L²)≡{g | g ∈ L²(0, T; D(Δ)), g^′ ∈ L²(Q)}, and the embedding D(Δ)↪L² is compact, we can employ the Aubin-Lions-Temam’s compact embedding theorem (cf. Temam [27, p. 274]) to determine that the embedding H¹(0, T; D(Δ), L²)↪L²(Q) is compact. Consequently, the observer C₂ is a compact operator. Therefore, C₂ satisfies the requirement for the existence of optimal pairs given in Theorem 11.

4.2.1. Case of Distributive and Terminal Values Observations C₁

In this observation case, we consider the cost function associated with the control system (68):

(104)

where Y_d ∈ L²(Q) and

are desired values, and the positive constants α and β are the relative weight of the second and the third terms on the RHS of (104).

Now we formulate the following adjoint equation to describe the necessary optimality conditions for this observation:

(105)

where

is defined in (33). Using a similar estimation to (34), we can have

(106)

Remark 14. By considering the observation conditions y(q^∗) − Y_d ∈ L²(Q) ⊂ L²(0, T; H⁻¹) and and (106), we can refer to the well-posedness result of Dautray and Lions [23, pp.558-570] to verify that (105), reversing the direction of time t → T − t, admits a unique weak solution p ∈ W(0, T), which is given in Definition 2.

We now discuss the first-order optimality conditions for the minimax optimal control problem (71) for the quadratic cost function (104).

Theorem 15. If α and β in the cost (104) are large enough, then an optimal control and a disturbance , namely, an optimal pair satisfying (71), can be given by

(107)

where p is the weak solution of (105).

Proof. Let be an optimal pair in (71) with the cost (104) and let y(q^∗) be the corresponding weak solution of (68).

From Theorem 7, we know that the map q = (u, v) → y(q) is Fréchet differentiable at q = q^∗ = (u^∗, v^∗) in the direction , which satisfies for sufficiently small ϵ > 0. Thus, the map q = (u, v) → y(q) is also (strongly) Gâteaux differentiable at q = q^∗ in the direction . Thus, we have

(108)

where z = Dy(q^∗)w is a unique solution of (69). Therefore we can obtain the Gâteaux derivative of the cost (104) at q = q^∗ in the direction w = (h, l) as follows:

(109)

where z = Dy(q^∗)w is a solution of (69).

Before we proceed to the calculations, we note that

(110)

We multiply both sides of the weak form of (105) by z, which is a solution of (69), and integrate it over [0, T]. Then, we have

(111)

By integration by parts and the terminal value of the weak solution p of (105), (111) can be rewritten as

(112)

Since z is the solution of (69), we can obtain the following from (112):

(113)

Therefore, we can deduce that (109) and (113) imply

(114)

Since

is an optimal pair in (71), we know that

(115)

Therefore, we can obtain the following from (114) and (115):

(116)

where

By considering the signs of the variations h and l in (116), which depend on u^∗ and v^∗, respectively, we can deduce the following from (116) (possibly not unique):

(117)

This completes the proof.

4.2.2. Case of Velocity Observation C₂

In this observation case, we consider the cost function associated with the control system (68):

(118)

where Y_d ∈ L²(Q) is a desired value and the positive constants α and β are the relative weight of the second and the third terms on the RHS of (118). Now we turn to the necessary optimality conditions that have to be satisfied by each solution of the minimax optimal control problem with the cost (118). For this purpose, as proposed in a previous study [8], we introduce the following adjoint equation corresponding to (68), in which q = (u, v) is replaced by q^∗ = (u^∗, v^∗):

(119)

where

is defined in (33).

Remark 16. Usually, adjoint systems of second order problems are also second order (cf. Lions [9]) as long as they are meaningful. However, we have a barrier in this quasilinear (68). If we derive a formal second order adjoint system related to the velocity observation with the cost (118), then it is hard to explain the well-posedness. To overcome this difficulty, we follow the idea given in [8, 11], in which it is adopted that the first-order integrodifferential system as an appropriate adjoint system instead of the formal second order adjoint system.

Proposition 17. Equation (119) admits a unique weak solution p satisfying

(120)

where

is the solution space of (119) given by

(121)

Proof. Since

(122)

the time reversed equation of (119) (t → T − t in (119)) is given by

(123)

where ψ(·) = p(T − ·). From (106) and −y^′(q^∗) − Y_d ∈ L²(Q), it is verified that all requirements of Dautray and Lions [23, pp.656-661] are satisfied with (123). Therefore, it readily follows that there exists a unique weak solution

of (123).

This completes the proof.

We now discuss the first-order optimality conditions for the minimax optimal control problem (71).

Theorem 18. If α and β in the cost (118) are large enough, then an optimal control and a disturbance , namely, an optimal pair satisfying (71), can be given by:

(124)

where p is the weak solution of (119).

Proof. Let be an optimal pair in (71) with the cost (118) and y(q^∗) be the corresponding weak solution of (68).

By analogy with the proof of Theorem 15, the Gâteaux derivative of the cost (118) at q^∗ = (u^∗, v^∗) in the direction that satisfies for sufficiently small ϵ > 0 is given by

(125)

where z = Dy(q^∗)w is a solution of (69). We multiply both sides of the weak form of (119) by z^′ and integrate it over [0, T]. Then, we have

(126)

By integration by parts and the terminal value of the weak solution p of (119), (126) can be rewritten as

(127)

Since z is the solution of (69), we can obtain the following from (127):

(128)

Therefore, we can deduce that (125) and (128) imply

(129)

Since

is an optimal pair in (71), we know that

(130)

Therefore, we can obtain the following from (129) and (130):

(131)

where

By considering the signs of the variations h and l in (131), which depend on u^∗ and v^∗, respectively, we can deduce from (131) that (possibly not unique)

(132)

This completes the proof.

5. Conclusion

The Fréchet differentiability from a bilinear control input into the solution space of a damped Kirchhoff-type equation is verified. As an application of this result, we proposed a minimax optimal control problem for the above state equation by using quadratic cost functions that depend on control and disturbance (or noise) variables. By utilizing the Fréchet differentiability of the solution map and the continuity of the solution map in a weak topology, we have proven existence of the optimal control of the worst disturbance, called the optimal pair under some hypothesis. And we derived necessary optimality conditions that any optimal pairs must satisfy in some observation cases.

Conflicts of Interest

The author declares no conflicts of interest.

Authors’ Contributions

The author read and approved the final manuscript.

Acknowledgments

This research was supported by the Daegu University Research Grant 2015.

Open Research

Data Availability

No data were used to support this study.

References

1 Kirchhoff G., Vorlesungen über Mechanik, 1883, Teubner, Leipzig, Germany.
Google Scholar
2 Arosio A., Averaged evolution equations. The Kirchhoff string and its treatment in scales of Banach spaces, Proceedings of the nd Workshop on Functional-Analytic Methods in Complex Analysis, Treste, 1993, Singapore, World Scientific, MR1414220.
Google Scholar
3 Spagnolo S., The Cauchy problem for Kirchhoff equations, Rendiconti del Seminario Matematico e Fisico di Milano. (1992) 62, 17–51, MR1293773.
10.1007/BF02925435
Google Scholar
4 Pohozaev S., On a class of quasilinear hyperbolic equations, Matematicheskii Sbornik. (1975) 96, 152–166, MR0369938.
Google Scholar
5 Lions J. L., G. M. de la Penha and L. A. Medeiros, On some questions in boundary value problem of Mathematical Physics, Contemporary developments in Continuum Mechanics and Partial Differential Equations, 1977, Math. Studies, North Holland.
Google Scholar
6 Nishihara K. and Yamada Y., On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms, Funkcialaj Ekvacioj. (1990) 33, no. 1, 151–159, MR1065473.
Google Scholar
7 Cavalcanti M. M., Domingos Cavalcanti V. N., Prates Filho J. S., and Soriano J. A., Existence and exponential decay for a Kirchhoff-Carrier model with viscosity, Journal of Mathematical Analysis and Applications. (1998) 226, no. 1, 40–60, MR1646453, https://doi.org/10.1006/jmaa.1998.6057, Zbl0914.35081, 2-s2.0-0037539250.
10.1006/jmaa.1998.6057
Web of Science® Google Scholar
8 Hwang J.-s. and Nakagiri S.-i., Optimal control problems for Kirchhoff type equation with a damping term, Nonlinear Analysis, Theory, Method and Applications. (2010) 72, no. 3-4, 1621–1631, https://doi.org/10.1016/j.na.2009.09.002, MR2577563.
10.1016/j.na.2009.09.002
Web of Science® Google Scholar
9 Lions J. L., Optimal Control of Systems Governed by Partial Differential Equations, 1971, Springer-Verlag, Berlin, Germany, MR0271512.
10.1007/978-3-642-65024-6
Google Scholar
10 Droniou J. and Raymond J.-P., Optimal pointwise control of semilinear parabolic equations, Nonlinear Analysis. (2000) 39, 135–156, https://doi.org/10.1016/S0362-546X(98)00170-9, MR1722110.
10.1016/S0362-546X(98)00170-9
Web of Science® Google Scholar
11 Hwang J.-s. and Nakagiri S.-i., Optimal control problems for the equation of motion of membrane with strong viscosity, Journal of Mathematical Analysis and Applications. (2006) 321, no. 1, 327–342, https://doi.org/10.1016/j.jmaa.2005.07.015, MR2236562, 2-s2.0-33646361799.
10.1016/j.jmaa.2005.07.015
Web of Science® Google Scholar
12 Hwang J.-s., Optimal control problems for an extensible beam equation, Journal of Mathematical Analysis and Applications. (2009) 353, no. 1, 436–448, https://doi.org/10.1016/j.jmaa.2008.12.020, MR2508880, Zbl1170.35020, 2-s2.0-58149242318.
10.1016/j.jmaa.2008.12.020
Web of Science® Google Scholar
13 Belmiloudi A., Bilinear minimax control problems for a class of parabolic systems with applications to control of nuclear reactors, Journal of Mathematical Analysis and Applications. (2007) 327, no. 1, 620–642, https://doi.org/10.1016/j.jmaa.2006.04.037, MR2277438, Zbl1116.49014, 2-s2.0-33750804177.
10.1016/j.jmaa.2006.04.037
Web of Science® Google Scholar
14 Arada N. and Raymond J.-P., Minimax control of parabolic systems with state constraints, SIAM Journal on Control and Optimization. (1999) 38, no. 1, 254–271, https://doi.org/10.1137/S0363012998341411, MR1740597, Zbl0943.49020, 2-s2.0-0342757510.
10.1137/S0363012998341411
Web of Science® Google Scholar
15 Lasiecka I. and Triggiani R., Control theory for partial differential equations: continuous and approximation theories, I, 2000, Cambridge University Press, Cambridge, UK, MR1745475.
Google Scholar
16 Li X. and Yong J., Optimal Control Theory for Infinite Dimensional Systems, 1995, Birkhäuser, Boston, Mass, USA, https://doi.org/10.1007/978-1-4612-4260-4.
10.1007/978-1-4612-4260-4
Google Scholar
17 Barbu V. and Precupanu T., Convexity and optimization in Banach spaces, 1986, Reidel, Dordrecht, Netherlands, MR860772.
Google Scholar
18 Bradley M. E. and Lenhart S., Bilinear spatial control of the velocity term in a Kirchhoff plate equation, Electronic Journal of Differential Equations. (2001) 27, 1–15, MR1836795.
Google Scholar
19 Bradley M. E., Lenhart S., and Yong J., Bilinear optimal control of the velocity term in a Kirchhoff plate equation, Journal of Mathematical Analysis and Applications. (1999) 238, no. 2, 451–467, https://doi.org/10.1006/jmaa.1999.6524, MR1715493, Zbl0936.49003, 2-s2.0-0001534477.
10.1006/jmaa.1999.6524
Web of Science® Google Scholar
20 Lenhart S. and Liang M., Bilinear optimal control for a wave equation with viscous damping, Houston Journal of Mathematics. (2000) 3, no. 26, 575–595, MR1811943.
Google Scholar
21 Temam R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 1997, 68, 2nd edition, Springer, New York, NY, USA, Applied Mathematical Sciences, https://doi.org/10.1007/978-1-4612-0645-3, MR1441312, Zbl0871.35001.
10.1007/978-1-4612-0645-3
Google Scholar
22 Adams R. A., Sobolev Spaces, 1975, Academic Press, New York, NY, USA, MR0450957, Zbl0314.46030.
Google Scholar
23 Dautray R. and Lions J., Mathematical Analysis and Numerical Methods for Science and Technology, 2000, 5, Springer-Verlag, Evolution Problems I, https://doi.org/10.1007/978-3-642-58090-1.
Google Scholar
24 Lions J. L. and Magenes E., Non-Homogeneous Boundary Value Problems and Applications I, II, 1972, Springer-Verlag, Heidelberg, Germany, MR0350177.
Google Scholar
25 Evans L. C., Partial Differential Equations, 1998, 19, American Mathematical Society, Providence, RI, USA, Graduate Studies in Mathematics, MR1625845, Zbl0902.35002.
10.1090/gsm/019
Google Scholar
26 Simon J., Compact sets in the space L^p(0,T;B), Annali di Matematica Pura ed Applicata. (1987) 146, no. 4, 65–96.
10.1007/BF01762360
Web of Science® Google Scholar
27 Temam R., Navier-Stokes Equations Theory and Numerical Analysis, 1984, North-Holland, MR769654.
Google Scholar

All articles

Fréchet Differentiability for a Damped Kirchhoff-Type Equation and Its Application to Bilinear Minimax Optimal Control Problems

Abstract

1. Introduction

2. Preliminaries

3. Fréchet Differentiability of the Nonlinear Solution Map

4. Quadratic Cost Minimax Control Problems

4.1. Existence of Optimal Pairs

4.2. Necessary Conditions of Optimal Pairs

4.2.1. Case of Distributive and Terminal Values Observations C₁

4.2.2. Case of Velocity Observation C₂

5. Conclusion

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Fréchet Differentiability for a Damped Kirchhoff-Type Equation and Its Application to Bilinear Minimax Optimal Control Problems

Abstract

1. Introduction

2. Preliminaries

3. Fréchet Differentiability of the Nonlinear Solution Map

4. Quadratic Cost Minimax Control Problems

4.1. Existence of Optimal Pairs

4.2. Necessary Conditions of Optimal Pairs

4.2.1. Case of Distributive and Terminal Values Observations C1

4.2.2. Case of Velocity Observation C2

5. Conclusion

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

References

Related

Information

4.2.1. Case of Distributive and Terminal Values Observations C₁

4.2.2. Case of Velocity Observation C₂