# Necessary and Sufficient Conditions of Optimality for a Damped Hyperbolic Equation in One-Space Dimension

Authors: Ismail Kucuk; Kenan Yildirim
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The present paper deals with the necessary optimality condition for a class of distributed parameter systems in which the system is modeled in one-space dimension by a hyperbolic partial differential equation subject to the damping and mixed constraints on state and controls. Pontryagin maximum principle is derived to be a necessary condition for the controls of such systems to be optimal. With the aid of some convexity assumptions on the constraint functions, it is obtained that the maximum principle is also a sufficient condition for the optimality.


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## Body

## 1. Introduction

It is well known that many processes in science and engineering are modeled by partial differential equations. The problems concerning the control of vibrating systems are generally governed by hyperbolic partial differential equations which are obtained by using conservation laws as a description of the distributed parameter system. In order to control these systems, the derivation of necessary conditions in the form of a maximum principle has been studied since the 1960s.The Maximum principle is introduced for the first time by Pontryagin and his students as a necessary condition for the optimality of a mechanical system which is defined by ordinary differential or difference-differential equations [1]. Pontryagin’s maximum principle is given in the form of a Hamiltonian that is defined in terms of an adjoint variable and the control function. The first application of maximum principle was the maximization of the terminal velocity of a rocket. Following this initial work, it is shown that the maximum principle is a necessary condition, if the value of the Hamiltonian maximized over the controls is concave in the state variables and the sufficient condition with appropriate transversality conditions [2]. Necessary conditions of optimality for distributed parameter systems described by boundary-value problems for hyperbolic and parabolic equations are studied in [3] where the completeness assumption on the class of admissible controls is imposed. In [4], Necessary condition for optimality is derived in a form similar to Pontryagin’s maximum principle without admissible control function set which is bounded or closed. These necessary conditions are expressed in terms of certain “generalized Jocabians.” For a general control problem formulated in terms of a differential inclusion, maximum principle is given by weak pseudo-Lipschitz behavior that is postulated on the underlying multifunction [5]. Necessary condition in the form of Pontryagin maximum principle by adopting the Dubovitskiy-Milyutin functional analytical approach is derived in [6]. In [7], Barnes presents a maximum principle as a necessary condition for optimal control of vibrating system that is modeled by second order linear hyperbolic PDE where completeness assumption is dropped and a regular point is used. Furthermore, Barnes shows that, by certain convexity assumptions on the constraint functions, the maximum principle is also a sufficient condition for the optimality of distributed parameters systems. Other related theoretical studies about the maximum principle in the literature are available such as [8–25]. In [1–7], the optimal necessary and sufficient conditions are derived for a single partial differential equation without damping term subject to the homogeneous boundary conditions in one space dimension. The systems that involve only one control function are studied in [1–25], while studies in [26–30] examine the systems with multiple control functions due to the size of the structures or to increase the control efficiency.In the present paper, inspired by [7], the necessary optimality condition is given for a hyperbolic partial differential equation in one-space dimensional system. The system involves a damping and several control functions and is subject to the mixed integral constraints on state and control functions. The partial differential equation under consideration involves spatial derivatives of at most order four. The main goal of the control problem interested here is to minimize the performance index of the control problem in a given period of time with the control functions and state variable subject to the constraints in the form of integral equality or/and inequality. Under proper convexity assumptions on the constraint functions, the maximum principle is also a sufficient condition for a general class of hyperbolic partial differential equations in one-space dimensional system.


## 2. Mathematical Formulation of the Problem

Let us consider the following partial differential equation [31]:
(1)μ(x)wtt+N(x)[w]+M(x)[wt]=E(x,t)+∑j=1N‍fj(x,t)=E(x,t)+∑j=1N‍fj(x,t)on⁡Ω=(0,ℓ)×(0,tf),
where w(x,t) is the transversal displacement at (x,t)∈Ω¯={(x,t):x∈[0,ℓ],t∈[0,tf]}, x is the space variable, t is the time variable, E(x,t) is the external excitation, μ(x)>0 is the mass per unit length of the beam, tf is predetermined terminal time, and fj(x,t),  j=1,…,N are the control functions,
(2)M(x)[w]=∑j=04‍mj(x)(∂jw∂xj),N(x)[w]=∑j=04‍nj(x)(∂jw∂xj),
in which mj(x) and nj(x) are continuous functions on Ω, M(x) and N(x) are positive-definite operators, and M(x) is the damping operator. Equation (1) is subject to the following boundary conditions:(3a)ajk(x,t)∂jw∂xj|x=0=Tj+k(t),k=1,2,j=0,1,2,3(3b)ajk(x,t)∂jw∂xj|x=ℓ=Sj+k(t),k=3,4,j=0,1,2,3,where ajk(x,t), Sj+k(t), and Tj+k(t) are continuous functions on Ω and the initial conditions
(4)w(x,0)=w0(x)∈H1(0,ℓ),wt(x,0)=w1(x)∈L2(0,ℓ)
in which
(5)H1(0,ℓ)={w0(x)∈L2(0,ℓ):∂w0(x)∂x∈L2(0,ℓ)}.
The following assumptions are made:(A1)
∂jw/∂xj,  ∂j+1w/∂xj∂t, ∂2w/∂t2∈L2(Ω¯), j=0,1,…,4,Ω¯ is the closure of Ω;(A2)
ajk(x,t),  Tj+k(t),  Sj+k(t)∈L2(Ω)  k=1,…,4,  j=0,…,3;(A3)
the set of admissible control functions is given by(6)Uad={fk(x,t)∣fk(x,t)∈L2(Ω),|fk(x,t)|≤Mk<∞,k=1,…,Nfk(x,t)∣fk(x,t)∈L2(Ω)}.
in which L2(Ω) denotes the Hilbert space of real-valued square-integrable functions on the domain Ω in the Lebesgue sense with the usual inner product and norm defined by
(7)〈f,g〉Ω=∫Ω‍f(x,t)g(x,t)dΩ,∥f∥2=〈f,f〉,
respectively. Under these assumptions, the system equations (1)–(4) have a solution [32].


## 3. Formulation of the Control Problem

The optimal control problem aims to determine optimal control functionsfi∘(x,t)∈Uad,i=1,…,N that minimizes the performance index at the terminal time tf:
(8)𝒥0(f1(x,t),f2(x,t),…,fN(x,t))=∫0ℓ‍[𝒢1(x,w(x,tf))+𝒢2(x,wt(x,tf))]dx+∫0tf‍∫0ℓ‍𝒢0(x,t,w(x,t),f1(x,t),…,fN(x,t))dxdt.
The first two terms in the right hand side of (8) denote the modified energy of the system and the last term represents the control effort spent in control duration. kth admissible control function fk∘(x,t) subject to (1)–(4) and the following constraints:(9a)∫0ℓ‍h2k(x,wt(x,tf))dx+∫0tf‍∫0ℓ‍𝒢-2k(x,t,w(x,t),fk(x,t))dxdt=c-2k(9b)∫0ℓ‍h1k(x,w(x,tf))dx+∫0tf‍∫0ℓ‍𝒢-1k(x,t,w(x,t),fk(x,t))dxdt=c-1k(9c)∫0tf‍∫0ℓ‍𝒢ik(x,t,w(x,t),fk(x,t))dxdt≤cik,1≤i≤m(9d)∫0tf‍∫0ℓ‍𝒢ik(x,t,w(x,t),fk(x,t))dxdt=cik,m<i≤M,in which h1k, h2k, 𝒢0, 𝒢1, 𝒢2, and 𝒢ik, for k=1,2,…,N,   i=-2,-1,1,…,M, are continuous functions of their parameters. Also, h1k, 𝒢0, 𝒢1, and 𝒢ik, for k=1,2,…,N,  i=-2,-1,1,…,M, have the continuous derivatives with respect to w and h2k, 𝒢2 have the continuous derivatives with respect to wt.


## 4. Necessary and Sufficient Conditions for Optimality

In this section, necessary condition of optimality is derived in the form of the maximum principle. Also, under proper convexity assumptions on the constraint functions, it is shown that maximum principle is also sufficient condition of optimality. For convenience, let us assume that(10)𝒢0(x,t,w,f1(x,t),…,fN(x,t))=∑k=1N‍𝒢0k(x,t,w,fk(x,t)),
where 𝒢0k(x,t,w,fk(x,t)) is the related term with kth control function in 𝒢0. In order to achieve the maximum principle, an adjoint variable v(x,t) along with the adjoint operator is introduced. The adjoint variable v(x,t) satisfies the following equation:
(11)μ(x)vtt+N*(x)[v]-M*(x)[vt]=∑k=1N‍∑i=-2M‍λik∂𝒢ik∂w(x,t,w,fk(x,t)),λ0k=λ0,
where we introduced the Lagrange multiplier λik≤0 and
(12)M*(x)[v]=∑j=04‍(-1)j(∂j∂xj)(vmj(x)),N*(x)[v]=∑j=04‍(-1)j(∂j∂xj)(vnj(x)).
Equation (11) is subject to the following homogeneous boundary conditions:(13a)ajk(x,t)∂jv∂xj|x=0=0,k=1,2,j=0,1,2,3(13b)ajk(x,t)∂jv∂xj|x=ℓ=0,k=3,4,j=0,1,2,3and the terminal conditions(14a)v(x,tf)=1μ(x)λ0∂𝒢2∂wt(x,wt(x,tf))(x,tf)=+1μ(x)∑k=1N‍λ-2k∂h2k∂wt(x,wt(x,tf)),(14b)vt(x,tf)-M*(x)[v](x,tf)=-1μ(x)∑k=1N‍λ-1k∂h1k∂w(x,w(x,tf))-1μ(x)λ0∂𝒢1∂w(x,w(x,tf)).Let us introduce a special perturbation of the optimal control problem and three lemmas to derive the maximum principle. Suppose thatfk∘(x,t),   k=1,…,N are optimal control functions corresponding to optimal displacement w∘. Let (x1,t1),…,(xP,tP) be P arbitrary points in the open region Ω and let fkj(x,t),   k=1,…,N,  j=1,…,P be P arbitrary subfunctions for every admissible control function fk∈Uad,  k=1,2,…,N. Also, let us assume that x1≤x2≤⋯≤xP. Choose ς>0 such that xi+Pς<xj, if xi<xj,   xP+Pς<ℓ and ti+ς<tf, for each 0≤i≤P. Let ε1,…,εP be real parameters satisfying 0≤εj≤ς2. Let X1=x1 and Xj=xj+ε1+⋯+εj-1 be for 1<j≤P. Hence, the intervals Xj≤x≤Xj+εj and the rectangles Rj:[Xj,Xj+εj]×[tj,tj+εj] do not have any intersection for 1≤j≤P, respectively. ɛ denotes the vector (ε1,…,εP)∈ℝP; ℝP is P-dimensional Euclidean space, and the norm of ɛ is given by |ɛ|=ε1+⋯+εP. The admissible controls fkɛ(x,t)∈Ω¯ are defined by
(15)fkɛ(x,t)={fk∘(x,t),if(x,t)∉⋃j=1P‍Rj,fkj(x,t),if(x,t)∈Rj,j=1,…,P,
for k=1,2,…,N.Lemma 1.
Letwɛ satisfy the system given by (1)–(4) corresponding to the controls f1ɛ(x,t),f2ɛ(x,t),…,fNɛ(x,t). Consider the following difference functions,
(16)ΔF(x,t)=Fɛ(x,t)-F∘(x,t),Δw(x,t)=wɛ(x,t)-w∘(x,t),
in which
(17)F(x,t)=∑k=1N‍fk(x,t).
Note that Δw(x,t) satisfies following equation:
(18)μ(x)Δwtt+N(x)[Δw]+M(x)[Δwt]=ΔF(x,t),and the following homogeneous boundary conditions:(19a)bjk(x,t)∂jΔw∂xj|x=0=0,k=1,2,j=0,1,2,3(19b)bjk(x,t)∂jΔw∂xj|x=ℓ=0,k=3,4,j=0,1,2,3and also, the zero initial conditions:
(20)Δw(x,0)=0,Δwt(x,0)=0.
Then,
(21)∫0ℓ‍Δw2(x,tf)dx=o(ɛ),∫0ℓ‍Δwt2(x,tf)dx=o(ɛ),∫0tf‍∫0ℓ‍Δw2(x,t)dxdt=o(ɛ).o(ɛ) is a quantity such that
(22)lim⁡ɛ→0(o(ɛ)|ɛ|)=0.Proof.
We start the proof by examining the following energy integral:(23)E(t)=12∫0ℓ‍{(μ(x)Δwt)2+12N(x)[Δw]2}dx;
then, it can be rewritten in [33] as(24)E(t)=∫0t‍dE(τ)dτdτ=∫0t‍12∫0ℓ‍{12N(x)[2ΔwΔwt]2μ(x)ΔwtΔwtt=∫0t‍12∫0ℓ‍+12N(x)[2ΔwΔwt]}dxdτ.
With integration by parts and using homogeneous boundary conditions given by (19a) and (19b), (24) becomes
(25)E(t)=∫0t‍∫0ℓ‍{μ(x)Δwtt+N(x)[Δw]}Δwtdxdτ=∫0t‍∫0ℓ‍{ΔF(x,τ)Δwt-ΔwtM(x)[Δwt]}dxdτ≤∫0t‍∫0ℓ‍ΔF(x,τ)Δwt(x,τ)dxdτ.
Note that if N(x) is nonself adjoint operator, the foregoing inequality is also satisfied. By applying the Cauchy-Schwartz inequality to the space integral, we obtain
(26)E(t)≤∫0t‍[∫0ℓ‍(Δwt)2dx]1/2[∫0ℓ‍(ΔF(x,τ))2dx]1/2dτ≤∫0t‍E1/2(τ)[∫0ℓ‍(ΔF(x,τ))2dx]1/2dτ.
Taking the sup of both sides of (26) leads to
(27)sup⁡E(t)≤sup⁡E1/2(t)∫0t‍[∫0ℓ‍(ΔF(x,τ))2dx]1/2dτ=sup⁡E12(t)∑i=1P‍o(εi5/4),
where o(r) is a quantity such that
(28)lim⁡r→0+(o(r)r)=constant.
By means of (27), the following inequality is observed for each t∈[0,tf]:
(29)0≤E1/2(t)≤o(ε5/4).
Because 5/4>0 [34], the following equality is obtained:
(30)E(t)=o(ɛ).
Because the coefficients of (1) are bounded away from zero, the conclusion of Lemma 1 is obtained from (30). It is concluded from Lemma 1 that
(31)lim⁡ΔF(x,t)→0Δw(x,t)=0.
Namely, (1)–(4) have a unique solution.Let us define the differential operatorℒ and its adjoint operator ℳ as follows:
(32)ℒ[w]≡μ(x)wtt+N(x)[w]+M(x)[wt],ℳ[v]≡μ(x)vtt+N*(x)[v]-M*(x)[vt].Lemma 2.
Letv and Δw(x,t)=w(x,t)-w∘(x,t) be two functions which are defined in L2(Ω). Also, let us assume that v and Δw(x,t) satisfy conditions (13a) and (13b) and (19a) and (19b)-(20), respectively. Then,
(33)∫0tf‍∫0ℓ‍{vℒ[Δw]-Δwℳ[v]}dxdt=∫0ℓ‍μ(x){-Δw(x,tf){vt(x,tf)-M*(x)[v(x,tf)]}v(x,tf)Δwt(x,tf)-Δw(x,tf)=∫0ℓ‍μ(x)×{vt(x,tf)-M*(x)[v(x,tf)]}}dx.Proof.
The reader is referred to [31].Definition 3.
For the arbitrary constants,λ-2k,λ-1k,λ0,…,λMk, k=1,…,N, v is the solution of the system equations (11)–(14a) and (14b). Let w∘ be the response function corresponding to optimal control functions fk∘∈Uad, k=1,…,N,  u be any of the following functions:
(34)u(x,t)=𝒢ik(x,t,w,fk(x,t)),fk∈Uadisfixedu(x,t)=𝒢ik(x,t,w∘,fk∘(x,t)),u(x,t)=∑k=1N‍vfk∘,fk∘∈Uad,u(x,t)=∑k=1N‍vfk,fk∈Uadisfixed.
A point (x¯,t¯) is called a regular point, for each fk,k=1,…,N∈Uad, if it satisfies the following equality for any sufficiently small ɛ>0:
(35)∫t¯t¯+ɛ‍∫x¯x¯+ɛ‍u(x,t)dxdt=ɛu(x¯,t¯)+o(ɛ).
It can be concluded from [35] that all points of [0,ℓ]×[0,tf] are regular for each fk,k=1,…,N∈Uad.Let𝒥k and Z denote the vector valued functional (𝒥-2k,𝒥-1k,𝒥0k,…,𝒥Mk), for k=1,…,N, and the set
(36)Z={𝒥k(fk):fk∈Uad}⊂ℝM+3.Definition 4.
If there exists a surface in the following form:(37)𝒥kɛ=𝒥k(fk∘)+∑j=1P‍dkjεj+o(ɛ)
in Z for sufficiently small εj and dk1,…,dkP are any finite collection of vectors from D, then the set D is called as a derived set of the set Z at 𝒥k(fk∘) [36].Lemma 5.
Assume that the points(xi,ti) are regular points in Ω, for i=1,2,…,P, and 𝒥¯k(fk) is introduced as follows, for any fk∈Uad, k=1,…,N:
(38)𝒥¯k(fk)=∫0ℓ‍{(λ0N)∫0ℓ‍×[𝒢1(x,w(x,tf))+𝒢2(x,wt(x,tf))]∫0ℓ‍+[(x,w(x,tf))λ-2kh2k(x,wt(x,tf))∫0ℓ‍++λ-1kh1k(x,w(x,tf))](λ0N)[𝒢1(x,w(x,tf))+𝒢2(x,wt(x,tf))]}dx+∫0tf‍∫0ℓ‍∑i=-2M‍[λik𝒢ik(x,t,w,fk)]dxdt.
IfP=1, for k=1,…,Nfk∈Uad, there exist constants λ-2k,λ-1k,λ0,…,λMk (not all zero) such that
(39)λ0≤0,λik≤0(0≤i≤m),lim⁡ɛ→0+𝒥¯k(fkɛ)-𝒥¯k(fk∘)ɛ≤0,
in which fk∘’s and fɛk’s are functions which are defined in (15).Proof.
Define the functionals𝒥-2k,𝒥-1k,𝒥1k,…,𝒥Mk on the class of admissible controls by
(40)𝒥-2k(fk)=∫0ℓ‍h2k(x,wt(x,tf))dx+∫0tf‍∫0ℓ‍G-2k(x,t,w,fk)dxdt,𝒥-1k(fk)=∫0ℓ‍h1k(x,wt(x,tf))dx+∫0tf‍∫0ℓ‍G-1k(x,t,w,fk)dxdt,𝒥ik(fk)=∫0tf‍∫0ℓ‍Gik(x,t,w,fk)dxdt,i=1,…,M.
In order to use the Lagrange multiplier rule, we should construct a derived setD for the set Z at 𝒥k(fk∘) [36]. To this end, functions vjk are introduced, for k=1,2,…,N,   j=-2,-1,1,…,M, satisfying(41a)μ(x)∂2vjk∂t2+N*(x)[vjk]-M*(x)[∂vjk∂t]=∂𝒢jk∂w(x,t,w,fk),0≤x≤ℓ,0≤t≤tf,(41b)aiα(x,t)∂ivjk∂xi|x=0=0,α=1,2,i=0,1,2,3-2≤j≤M,k=1,…,N,(41c)aiα(x,t)∂ivjk∂xi|x=ℓ=0,α=3,4,i=0,1,2,3-2≤j≤M,k=1,…,N,(41d)v-2k(x,tf)=1μ(x)∂h2k∂wt(x,wt(x,tf)),∂v-2k∂t(x,tf)=0,(41e)v-1k(x,tf)=0,∂v-1k∂t(x,tf)=-1μ(x)∂h1k∂w(x,w(x,tf)),(41f)vjk(x,tf)=0,∂vjk∂t(x,tf)=0j=1,…,M,k=1,…,Nand, for k=1,…,N and j=0, satisfying (42a)μ(x)∂2v0k∂t2+N*(x)[v0k]-M*(x)[∂v0k∂t]=1N∂𝒢0∂w(x,t,w,fk),(42b)v0k(x,tf)=1Nμ(x)∂𝒢2∂wt(x,wt(x,tf))(42c)∂v0k∂t(x,tf)-M*(x)[v0k](x,tf)=-1Nμ(x)∂𝒢1∂w(x,w(x,tf)).For each point, (x,t)∈(0,ℓ)×(0,tf),i=-2,-1,0,…,M,k=1,…,N,dki(x,t,f¯k) is defined as follows:
(43)dki(x,t,f¯k)=vik(x,t)(f¯k-fk∘)+𝒢ik(x,t,w∘(x,t),f¯k)-𝒢ik(x,t,w∘(x,t),fk∘).
To show that
(44)D={dk∣dk=(dk-2(x,t,f¯k),…,dkM(x,t,f¯k)),(x,t)aregularpointoffk∘,f¯k∈Uaddk∣dk=(dk-2(x,t,f¯k),…,dkM(x,t,f¯k)),}
is a derived set for Z at 𝒥k(fk∘), let dk1,dk2,…,dkP be an arbitrary finite collection of vectors from D. We must show that there exist points 𝒥ɛk∈Z, which are subject to the vector parameter ɛ=(ε1,…,εP) for all sufficiently small positive values of ɛ such that
(45)𝒥ɛk=𝒥k(fk∘)+∑j=1P‍dkjεj+o(ɛ).
Since dkj∈D,  j=1,…,Pk=1,…,N, there exist (x1,t1),…,(xP,tP) regularity points of fk∘ and subfunctions fk1,…,fkP∈Uad such that
(46)dkj=(dk-2(xj,tj,fkj),…,dkM(xj,tj,fkj)),j=1,…,P,k=1,…,N.
To show that 𝒥ɛk can be written as 𝒥ɛk=𝒥k(fkɛ) where fkɛ is the admissible control given by (15), for i=1,…,M, we observe that
(47)𝒥ik(fkɛ)-𝒥ik(fk∘)=∫0tf‍∫0ℓ‍[-𝒢ik(x,t,w∘(x,t),fk∘(x,t))𝒢ik(x,t,wɛ(x,t),fkɛ(x,t))=∫0tf‍∫0ℓ‍-𝒢ik(x,t,w∘(x,t),fk∘(x,t))]dxdt=∫0tf‍∫0ℓ‍[𝒢ik(x,t,wɛ(x,t),fkɛ(x,t))=∫0tf‍∫0ℓ‍-𝒢ik(x,t,w∘(x,t),fkɛ(x,t))]dxdt+∫0tf‍∫0ℓ‍[𝒢ik(x,t,w∘(x,t),fkɛ(x,t))+∫0tf‍∫0ℓ‍-𝒢ik(x,t,w∘(x,t),fk∘(x,t))]dxdt=∫0tf‍∫0ℓ‍∂𝒢ik∂w(x,t,w∘,fk∘(x,t))Δw(x,t)dxdt+∑j=1P‍εj[-𝒢ik(xj,tj,w∘(xj,tj),fkj∘(xj,tj))𝒢ik(xj,tj,w∘(xj,tj),fkj)+∑j=1P‍εj-𝒢ik(xj,tj,w∘(xj,tj),fkj∘(xj,tj))]+∑j=1P‍o(εj).
In order to obtain (47), we use that fk∘ is regular at each of the points (xj,tj) and we use the conclusion of Lemma 1. If the following equality is substituted in (33)
(48)ℳvki=∂𝒢ik∂w(x,t,w∘(x,t),fk∘),i=1,…,M,k=1,…,N,
it is observed that
(49)∫0ℓ‍∫0tf‍Δw(x,t)ℳvkidxdt=∫0ℓ‍∫0tf‍vki(x,t)(fkɛ(x,t)-fk∘(x,t))=∑j=1P‍εjvki(xj,tj)(fkj-fkj∘(xj,tj))+o(ɛ).
By means of (43) and (47), we can write
(50)𝒥ik(fkɛ)=𝒥ik(fk∘)+∑j=1P‍dkjiεj+o(ɛ),i=1,…,M,k=1,…,N,
where dkji denotes the ith component of dkj. For i=0, we have
(51)𝒥0k(fkɛ)-𝒥0k(fk∘)=∫0ℓ‍[1N∂𝒢1∂w(x,w∘(x,tf))Δw(x,tf)=∫0ℓ‍+1N∂𝒢2∂wt(x,wt∘(x,tf))Δwt(x,tf)]dx+∑j=1P‍εj[𝒢0k(xj,tj,wɛ(xj,tj),fkj)+∑j=1P‍εj-𝒢0k(xj,tj,wɛ(xj,tj),fkj∘)]+∫0ℓ‍∫0tf‍(ℳv0k)Δw(x,t)dxdt+o(ɛ),
in which 𝒥0k=(1/N)∫0ℓ[𝒢1(x,w(x,tf))+𝒢2(x,wt(x,tf))]dx+∫0tf∫0ℓ𝒢0k(x,t,w,fk)dxdt. Considering (33) and (41a), (41b), (41c), (41d), (41e), and (41f), it is observed that
(52)∫0tf‍∫0ℓ‍Δw(x,t)ℳv0kdxdt=∫0tf‍∫0ℓ‍v0k(x,t)(fkɛ(x,t)-fk∘(x,t))dxdt-∫0ℓ‍[1N∂𝒢1∂w(x,w∘(x,tf))Δw(x,tf)-∫0ℓ‍+1N∂𝒢2∂wt(x,wt(x,tf))Δwt(x,tf)].
If (52) is substituted into (51), (50) is obtained for i=0,  k=1,…,N. For i=-2,-1, (50) can be obtained by using (33)–(41a), (41b), (41c), (41d), (41e), and (41f). By the definition 𝒥k, one obtains
(53)𝒥k(fkɛ)-𝒥k(fk∘)=∑j=1P‍dkjεj+o(ɛ),
where 𝒥kɛ is taken as 𝒥k(fkɛ). This completes the proof of D being a derived set for Z at 𝒥k(fk∘). Then, there exist nonpositive Lagrange multipliers [36], not zero simultaneously, satisfying
(54)∑i=-2M‍λikdki≤0,λ0k=λ0,
for any vector dk=(dk-2,dk-1,dk0,…,dkM) in D. Let us take P=1 in the foregoing discussion to obtain the conclusion of Lemma 5 and put
(55)𝒥¯k=∑i=-2M‍λik𝒥ik.
By (50), it follows that
(56)𝒥¯k(fkɛ)-𝒥¯k(fk∘)=ɛ∑i=-2M‍λikdki+o(ɛ),
for any dk=(dk-2,dk-1,dk0,…,dkM) in D. Then, we obtain the proof of Lemma 5 as follows:
(57)lim⁡ɛ→0+𝒥¯k(fkɛ)-𝒥¯k(fk∘)ɛ=∑i=-2M‍λikdki≤0.Theorem 6 (maximum principle).
For the optimal control functionsf1∘(x,t),…,fN∘(x,t)∈Uad, the corresponding optimal state and adjoint variables arew∘(x,t)=w(x,t,f1∘(x,t),f2∘(x,t),…,fN∘(x,t)) satisfying (1)–(4) and v∘(x,t)=v(x,t,f1∘(x,t),f2∘(x,t),…,fN∘(x,t)) satisfying (11), boundary conditions (13a) and (13b), and terminal conditions (14a) and (14b), respectively. The maximum principle states that if
(58)ℋ[x,t,v∘,f1∘(x,t),…,fN∘(x,t)]=max⁡fk∈Uadℋ[x,t,v,f1(x,t),…,fN(x,t)],k=1,…,N,
where the Hamiltonian is given by
(59)ℋ[x,t,v,f1,…,fN]=∑k=1N‍v(x,t)fk+∑k=1N‍∑i=-2M‍λik𝒢ik(x,t,w(x,t),fk),
then the performance index equation (8) is minimized; that is,
(60)𝒥0[f1∘(x,t),…,fN∘(x,t)]≤𝒥0[f1(x,t),…,fN(x,t)]foranyfk∈Uad.Proof.
Let(x,t) be a regular point for admissible optimal control functions fk∘,k=1,…,N. By Lemma 5, for 0≤i≤m and some λik≠0, there exists Lagrange multipliers λ-2k,λ-1k,λ0,…,λNM independent of (x,t) with λik≤0 such that
(61)∑k=1N‍∑i=-2M‍λik[vik(x,t)(fk-fk∘(x,t))+𝒢ik(x,t,w(x,t),fk)-𝒢ik(x,t,w∘(x,t),fk∘(x,t))]≤0,
for any function fk∈Uad. Note that the term
(62)∑k=1N‍∑i=-2M‍λikvik(x,t)fk+∑k=1N‍∑i=-2M‍λik𝒢ik(x,t,w(x,t),fk)λikvik(x,t)fk+∑k=1N‍∑i=-2M‍λik𝒢ik(x,t,w(x,t),fk)in(61)
reaches its maximum value at fk=fk∘(x,t)∈Uad, for k=1,…,N. Let us consider the first term in (62),
(63)∑k=1N‍∑i=-2M‍λikvik(x,t)fk,
that can be rewritten in the form
(64)∑k=1N‍∑i=-2M‍∑r=1N‍λikvik(x,t)frto be subject to
(65)λikvik(x,t)fr={λikvik(x,t)fk,r=k,0,r≠k.
If we define v=∑k=1N∑i=-2Mλikvik(x,t), we obtain
(66)∑k=1N‍v(x,t)fk+∑k=1N‍∑i=-2M‍λik𝒢ik(x,t,w(x,t),fk),
the conclusion of the maximum principle. This completes the proof of Theorem 6.Theorem 7.
Consider the control system equations (1)–(4) and (8)-(9a), (9b), (9c), and (9d). Let the functions Gik be in the form
(67)Gik(x,t,w,fk)=𝒢ki(x,t,w)+Hki(x,t,fk),i=-2,-1,0,…,M,k=1,…,N
and let v satisfying (13a) and (13b)-(14a) and (14b) be the nonzero solution of
(68)ℳv=∑k=1N‍∑i=-2M‍λik∂Gki(x,t,w∘(x,t))∂w.
Assume that there exist admissible control functions f1∘,f2∘,…,fN∘ and the constants λ0, λik, i=-2,-1,1,…,M, k=1,…,N that satisfy the maximum principle equations (58). Let us assume that following assumptions are satisfied:(a)
fork=1,…,N,  𝒢1,h1k,𝒢ik,…,𝒢mk are convex functions of w and 𝒢2,  h2k are convex functions of wt;(b)
λ0<0,   λik≤0, for i=-1,…,m,  k=1,…,N;(c)
the constraints equation (9a), (9b), (9c), and (9d) are satisfied by f1∘,f2∘,…,fN∘;(d)
if the strict inequality holds in (9a), (9b), (9c), and (9d), the corresponding Lagrange multiplier λik=0;(e)
-λikGki,  -λ-1kh1k are convex functions of w and -λ-2kh2k is convex function of wt, for k=1,…,N,m<i<M.
Under these assumptions, the maximum principle given by (58) is also sufficient condition for the admissible control functions f1∘,f2∘,…,fN∘ to be optimal. The condition (d) is proved in [36]. If the h1k,  Gik, and h2k are linear functions of w and wt, respectively, the condition (e) is satisfied, for k=1,…,N,  m<i≤M.Proof.
Iffk’s and w satisfy (9a), (9b), (9c), and (9d), then, by the condition (d),
(69)∫0tf‍∫0ℓ‍λik[Gki(x,t,w)-Gki(x,t,w∘)]dxdt+∫0tf‍∫0ℓ‍λik[Hki(x,t,fk)-Hki(x,t,fk∘)]dxdt=0,
for i=-2,-1,…,M,  k=1,…,N. Then, we can write the following inequality
(70)-λ0[𝒥0(f1,f2,…,fN)-𝒥0(f1∘,f2∘,…,fN∘)]≥-∫0ℓ‍λ0[𝒢2(x,wt(x,tf))-𝒢2(x,wt∘(x,tf))≥-∫0ℓ‍λ0+𝒢1(x,w(x,tf))-𝒢1(x,w∘(x,tf))]dx.-∫0tf‍∫0ℓ‍∑k=1N‍∑i=-2M‍λik{Gki(x,t,w(x,t))-Gki(x,t,w∘(x,t))-[Hki(x,t,fk(x,t))--Hki(x,t,fk∘(x,t))]}dxdt-∫0ℓ‍∑k=1N‍λ-1k[h1k(x,w(x,tf))-h1k(x,w∘(x,tf))]dx-∫0ℓ‍∑k=1N‍λ-2k[h2k(x,wt(x,tf))-h2k(x,wt∘(x,tf))]dx
By using the convexity assumption (e),
(71)-λ0[𝒥0(f1,f2,…,fN)-𝒥0(f1∘,f2∘,…,fN∘)]≥-∫0ℓ‍λ0[∂𝒢1∂w(x,w∘(x,tf)Δw(x,tf))≥-∫0ℓ‍λ0+∂𝒢2∂wt(x,wt∘(x,tf)Δwt(x,tf))]dx-∫0tf‍∫0ℓ‍∑k=1N‍∑i=-2M‍λik∂Gki∂w(x,t,w∘(x,t))Δw(x,t)dxdt+∫0tf‍∫0ℓ‍∑k=1N‍∑i=-2M‍λik[Hki(x,t,fk∘(x,t))+∫0tf‍∫0ℓ‍∑k=1N‍∑i=-2M‍λik-Hki(x,t,fk(x,t))]dxdt-∫0ℓ‍∑k=1N‍λ-1k∂h1k∂w(x,w∘(x,tf))Δw(x,tf)dx-∫0ℓ‍∑k=1N‍λ-2k∂h2k∂wt(x,wt∘(x,tf))Δwt(x,tf)dx=-∫0ℓ‍λ0[∂𝒢1∂w(x,w∘(x,tf))Δw(x,tf)=-∫0ℓ‍λ0+∂𝒢2∂wt(x,wt∘(x,tf))Δwt(x,tf)]dx-∫0tf‍∫0ℓ‍(ℳv)Δw(x,t)dxdt+∫0tf‍∫0ℓ‍∑k=1N‍∑i=-2M‍λik[Hki(x,t,fk∘(x,t))+∫0tf‍∫0ℓ‍∑k=1N‍∑i=-2M‍λik-Hki(x,t,fk(x,t))]dxdt-∫0ℓ‍∑k=1N‍[λ-1k∂h1k∂w(x,w∘(x,tf))Δw(x,tf)-∫0ℓ‍∑k=1N‍+λ-2k∂h2k∂wt(x,wt∘(x,tf))Δwt(x,tf)]dx.
And finally by applying Lemma 2 and the conditions (13a) and (13b)-(14a) and (14b), we obtain
(72)-λ0[𝒥0(f1,f2,…,fN)-𝒥0(f1∘,f2∘,…,fN∘)]≥∫0tf‍∫0ℓ‍{∑k=1N‍v(x,t)[fk∘(x,t)-fk(x,t)]≥∫0tf‍∫0ℓ‍+∑k=1N‍∑i=-2M‍λik[Hki(x,t,fk∘(x,t))≥∫0tf‍∫0ℓ‍+∑k=1N‍∑i=-2M‍λik-Hki(x,t,fk(x,t))]∑k=1N‍∑i=-2M‍}dxdt.
Note that the right-hand side of (72) is nonnegative due to the condition (b). Then, we obtain
(73)𝒥0(f1(x,t),f2(x,t),…,fN(x,t))-𝒥0(f1∘(x,t),f2∘(x,t),…,fN∘(x,t))≥0.
It follows that the maximum principle is also a sufficient condition for a global minimum of the performance index (8). It completes the proof of Theorem 7.


## 5. Conclusion

A necessary optimality condition for a general class of damped hyperbolic partial differential equation in one-space dimension is derived in the form of a maximum principle by using the derived set and regular point concepts. Under proper convexity assumption on state variable, it is proved that the maximum principle is also sufficient condition for the controls to be optimal. In [1–7], the systems under consideration include only one control function. Also, the necessary and sufficient conditions for optimality are given for a single hyperbolic differential equation without damping term subject to the homogeneous boundary conditions in one-space dimension. But, in the present paper, as an original contribution to literature, the necessary and sufficient optimality conditions obtained in [1–7] are generalized for a general class of damped hyperbolic equation involving damping and several control functions subject to nonhomogeneous boundary conditions in one space dimension.



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