# PID Controller Singularly Perturbing Impulsive Differential Equations and Optimal Control Problem

Authors: Wichai Witayakiattilerd
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

We study singular perturbation of impulsive system with a proportional-integral-derivative controller (PID controller) and solve an optimal control problem. The perturbation system comprises two important variables, a fast variable and a slow variable. Because of the complexity of the system, it is difficult to find its exact solution. This paper presents an approximation method for solving it. The aim of the approximation method is to reduce the complexity of the system by eliminating the fast variable. The solution of the method is expressed in an integral form, and it is called an approximated mild solution of the perturbed system. An example is provided to illustrate our result.


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## Body

## 1. Introduction

In this document, we tackle a combined problem of impulsive condition, singular perturbation and PID controller. They are introduced separately as follows.Impulsive differential equations play an important role in many areas of science such as physics, engineering, biology, and medicine. They can represent some natural phenomena better than ordinary differential equations, phenomena such as controlling chaos and bifurcation in engineering systems, epidemic model with impulsive birth, and modelling and control of complex dynamic systems ([1–16] and reference therein). Some of them are reviewed here. In 1989, Lakshmikantham et al. wrote an important book in this field and provided a theory named the theory of impulsive differential equations which have been cited by many researchers. In 2007, Jankowski investigated the existence of solutions and quasisolutions of advanced impulsive differential equations by applying Schauder’s fixed-point theorem. In 2009, Witayakiattilerd and Chonwerayuth proved the regularity of piecewise continuous almost periodic solutions for nonlinear impulsive systems by using contraction principle on Banach space. In 2012, Abdel-Rady et al. proved the existence and uniqueness of the solution of first-order impulsive differential equation and applied the results to second-order impulsive differential equation. In 2015, Li et al. derived the existence and global attractiveness of unique periodic solutions by using Lyapunov’s second method and the contraction mapping principle. In 2016, Li and Wu studied the stability of nonlinear differential systems with state-dependent delayed impulses. In 2017, Li and Cao presented a new impulsive delay inequality that involves unbounded and nondifferentiable time-varying delay.Some phenomena represented by an impulsive system may be perturbed by small parameters, in particular singularity perturbation. An impulsive differential equation with singularity perturbation is quite complicated and is difficult to find an exact solution, so there are not many researches in this field. Most of the researches were done by a few groups of mathematicians before year 2000 (see [3, 6, 9, 17–19]). For example, in 1992, Kulev studied the uniform asymptotic stability of impulsive perturbed systems of differential equations. In 1996, Bainov et al. investigated the uniform asymptotic stability of impulsive perturbed systems of differential equations. In 2007, Zhu et al. proved the exponential stability of singularly perturbed impulsive delay differential equations.A proportional-integral-derivative controller (PID controller) is, today, found in several areas where control has to be exerted. It is the most common form of feedback and is an important component of a distributed control system. It is always combined with logic, sequential functions, selectors, and simple function blocks for building complicated automation systems used for energy production, transportation, and manufacturing. Many sophisticated control strategies such as model predictive control are also organized hierarchically. For more details, please see ([20–23] and reference therein).The Abstract Cauchy Problem (ACP) of the singular perturbation of nonlinear functional impulsive differential equations on a Banach space of the problem is written as follows:(1)x˙t=Aεxt+ft,xt,yt,ε+ut,t≠tk,y˙t=1εBεyt+gt,xt,yt,ε,t≠tk,Δxtk=Ikxtk,ytk,Δytk=Jkxtk,ytk,x0=x0,y0=y0,where 0<ε≪1,  0<t1<t2<⋯<tn<T,  A(ε):X⊇D(A(ε))→R(A(ε))⊆X and B(ε):X⊇D(B(ε))→R(B(ε))⊆X are given continuous operators in ε,  Δx(tk)=x(tk+)-x(tk) and Δy(tk)=y(tk+)-y(tk) denote the jump of states x and y at time tk with the magnitude of jumps Ik and Jk, k=1,2,…, respectively. If f(·,·,·,ε) and g(·,·,·,ε) are globally Lipschitz and uniformly bounded in ε, then y˙ will be of order 1/ε faster than x˙. Consequently, we call x the slow variable and y the fast variable of the system. In this case, we assume that the system is controlled by PID controller:(2)ut=KPet+KI∫0teτdτ+KDdetdt,where e(t)=xsp(t)-x(t) and xsp(t) is reference variable called set point at time t. The constants Kp,  KI, and KD are nonnegative, called proportional gain, integral gain, and derivative gain, respectively.The complexity of the system makes it difficult to find its exact solution. This paper presents the approximation method for solving the system. The aim of the approximation method is to reduce the complexity of the system by eliminating the fast variable.The paper is organized as follows: in Section2, some basic concepts and notations which are useful for reading the next section are introduced; in Section 3, an approximated system is introduced and an approximate solution is presented; in Section 4, the existence of solution of optimal control problem is solved; an example is presented in Section 5.


## 2. Preliminaries

In this section, we state the definitions and theories that are used in this study. Throughout the paper,X denotes a Banach space with norm · and Y denotes another Banach space with norm ·Y. A linear transformation from X into Y is bounded on a domain of T,  D(T) if there exists a constant c such that TxY≤cx for all x∈D(T). The linear space of all bounded linear operators from X into Y is denoted by L(X,Y) and denote L(X,X) by L(X). In the proof of the existence and the uniqueness of solution, we use fixed-point theorem on Banach spaces or contraction mapping principle. An operator A:D(A)→R(A) is strictly contraction if there exists a constant c with 0≤c<1 such that Af1-Af2≤cf1-f2 for all f1,f2∈D(A). The contraction mapping principle states that if A:X→X is a strict contraction on Banach space X. Then the equation Af=f has a unique solution in X; that is, A has a unique fixed-point f. Furthermore, if X0 is a closed subset of the Banach space X and the strict contraction-operator A maps X0 into itself, then the equation Af=f has a unique solution f∈X0. Let Ttt≥0 be a one-parameter family of bounded linear operators from X into X such that Ttt≥0 is uniformly continuous, that is, limt→0+⁡T(t)-IL(X)=0.Definition 1.
A one-parameter familyTtt≥0 of bounded linear operators from X into X is a semigroup of bounded linear operators on X if (1)
T(0)=I, (I is the identity operator on X);(2)
T(t+s)=T(t)T(s) for every t,s∈R (the semigroup property).Define a linear operatorA by Ax=limt→0+⁡T(t)x-x/t=d+T(t)x/dt|t=0 for all x∈D(A) where D(A)={x∈X∣limt→0+⁡T(t)x-x/texists}. Then A is called the infinitesimal generator of a semigroup Ttt≥0. Definition 1 implies that for a semigroup {T(t)}t≥0 has a unique infinitesimal generator. If T(t) is uniformly continuous, its infinitesimal generator is a bounded operator. On the other hand, every bounded linear operator A is the infinitesimal generator of a uniformly continuous semigroup Ttt≥0 and this semigroup is unique. For more detail, please see [24].Definition 2.
A one-parameter familyTtt∈R of bounded linear operators from X into X is a group of bounded linear operators on X if (1)
T(0)=I, (I is the identity operator on X);(2)
T(t+s)=T(t)T(s) for every t,s∈R (the semigroup property);(3)
limt→0⁡T(t)x=x.Theorem 3.
LetA be an infinitesimal generator of the C0-semigroup {T(t)}t≥0. Then (1)
for allx∈X, limh→0+1/h∫tt+hTsxds=T(t)x;(2)
for allx∈X,∫0tT(s)xds∈D(A) and A∫0tT(s)xds=x-T(t)x;(3)
for allx∈D(A),T(t)x∈D(A),T(t)x∈D(A) and d/dtT(t)x=AT(t)x=T(t)Ax;(4)
for allx∈D(A),  T(s)x-T(t)x=∫stT(τ)Axdτ=∫stAT(τ)xdτ.


## 3. Main Results

In this section, we investigate the Abstract Cauchy Problem (ACP) of the singular perturbation of nonlinear functional impulsive differential equations on a BanachX, System (1). A suggested form of the system (1) is as follows:(3)x˙t=Aεxt+ft,xt,yt,ε+ut,t≠tk,εy˙t=Bεyt+gt,xt,yt,ε,t≠tk,Δxtk=Ikxtk,ytk,Δytk=Jkxtk,ytk,x0=x0,y0=y0with(4)ut=KPet+KI∫0teτdτ+KDdetdt, where e(t)=xsp(t)-x(t) and xsp(t) is a set point at time t. KP,  KI, and KD are constants.


### 3.1. Approximation Method

Suppose that(x0,y0) is arbitrary initial value. Then we can choose an initial (x0,y0) such that the fast equation becomes stationary, that is, y˙(t;y0)=0. This implies that(5)Bεy+gt,x,y,ε=0. Accordingly, for small enough ε, it seems reasonable to substitute (3) with a differential algebraic equation:(6)x˙=Aεxt+ft,xt,yt,ε+ut,t≠tk,0=Byt+Gt,xt,yt,t≠tk,Δxtk=Ikxtk,ytk,Δytk=Jkxtk,ytk,x0=x0,y0=y0,where B=B(0) and G(t,x(t),y(t))=g(t,x(t),y(t),0).Suppose thaty(t)=h(t,x(t))∈D(B)⊆X solves the following system:(7)0=Byt+Gt,xt,yt,t≠tk,Δytk=Jkxtk,ytk,y0=y0. Therefore, by substituting y(t)=h(t,x(t)) into (6), we have(8)x˙=Aε-KPI1+KDxt+11+KDft,xt,ht,xt,ε-KI∫0txτdτ+Xspt,t≠tk,Δxtk=Ikxtk,htk,xtk,x0=x0,where Xsp(t)=KPxsp(t)+KI∫0txspτdτ+KDx′sp(t). In other words, we can approximate the slow equation by the inhomogeneous ACP (8).First, we solve semilinear ACP:(9)x˙=Aε-KPI1+KDxt+11+KDft,xt,ht,xt,ε-KI∫t0txτdτ+Xspt,t0≤t≤θ,xt0=x0.Let Aad=u∣u(t)=KPe(t)+KI∫0te(τ)dτ+KDde(t)/dt,KP,KI,KD≥0,KD≠-1 be an admissible control set.Definition 4.
Letu∈Aad. A mild solution on [t0,θ] of the semilinear ACP (9) with respect to control u is a continuous function x:[t0,τ]→X such that x(t) satisfies the integral equation:(10)xt=Sεt-t0x0+11+KD∫t0tSεt-sfs,xs,hs,xs,ε-KI∫t0sxτdτ+Xspsds, where Sε(t)t≥0 is the C0-semigroup generated by the operator A^=A(ε)-KP/1+KD.We start with the following classical result which assures the existence and uniqueness of mild solutions of (9) for Lipschitz continuous function h and F.Theorem 5.
Letu∈Aad. Let h:[t0,θ]×X→X,  f·,·,·,ε:[t0,θ]×X×X→X be continuous in t on [t0,θ] and uniformly Lipschitz continuous (with constant Lh and LF,ε) on X. If A^=A(ε)-KPI/1+KD is infinitesimal generator of a C0-semigroup Sε(t)t≥0 on X, then for every x0∈X the semilinear ACP (9) has a unique mild solution x∈C([t0,θ],X) with respect to control u. Moreover, the mapping x0→x is Lipchitz continuous from X into C([t0,θ],X).Proof.
For a givenx0∈X. Define a mapping Q:C([t0,θ],X)→C([t0,θ],X) by(11)Qxt=Sεt-t0x0+11+KD∫t0tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xstsds.Denoting by x∞ the supremum norm of x as an element of C([t0,θ],X), it follows from definition of Q and the uniformly Lipschitz continuous of h and f that(12)Qx1t-Qx2t≤Mε1+KD∫t0tLF,εx1s-x2s+hs,x1s-hs,x2s+KI∫t0sx1τ-x2τdτds≤Mε1+KDLF,ε1+Lh+KIt-t0t-t0x1-x2∞≤Mε1+KDLF,ε1+Lhθ+KIθ2x1-x2∞,where Mε is a bound of Sε(t) on [t0,θ]. Using (11) and (12) and induction on n, we have(13)Qnx1t-Qnx2t≤Mε/1+KDLF,ε1+Lhθ+KIθ2nn!x1-x2∞. This implies that(14)Qnx1-Qnx2∞≤Mε/1+KDLF,ε1+Lhθ+KIθ2nn!x1-x2∞. For sufficiently large n,  Mε/1+KDLF,ε1+Lhθ+KIθ2n/n!<1 and by a well-known extension to the contraction mapping principle, there is a unique x∈C([t0,θ],X) such that(15)xt=Qxt=Sεt-t0x0+11+KD∫t0tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xstsds. Therefore, the semilinear ACP (9) has a unique mild solution x∈C([t0,θ],X). The Lipchitz continuity of the mapping x0→x is consequences of the following argument. Let x1 and x2 be the mild solutions of (9) with the initial values x10 and x20, respectively. Then it follows from the definition of mild solution and the uniformly Lipschitz continuous of h and f that(16)x1t-x2t≤Mεx10-x20+LF,ε1+Lh+KIC1+KD∫t0tx1s-x2sds for some constant C>0. By using Gronwall’s lemma, it can be implied that(17)x1-x2∞≤Mεexp⁡LF,ε1+Lh+KIC1+KDθx10-x20 which yields the Lipchitz continuity of the mapping x0→x.Corollary 6.
Under the conditions of Theorem5, for every φ∈C[t0,θ],X, the integral equation(18)λt=φt+11+KD∫t0tSεt-sfs,λs,hs,λs,ε-KI∫0sλτdτ+Xstsds has a unique solution λ∈C[t0,θ],X.Proof.
The proof is similar to the proof of Theorem5 by defining (Qλ)(t)=φ(t)+1/1+KD∫t0tSε(t-s)f(s,λ(s),hs,λs,ε)-KI∫0sλ(τ)dτ+Xst(s)ds.


### 3.2. Regularity of the System without Impulses

Definition 7.
Letu∈Aad. A classical solution of the inhomogeneous ACP (9) with respect to control u is a function x∈C[t0,θ],X such that (1)
x is continuously differentiable on (t0,θ);(2)
x(t)∈D(A) for all t∈[t0,θ];(3)
x satisfies the semilinear ACP (9).Theorem 8 (regularity).
Letu∈Aad and A^=A(ε)-KpI/1+KD be the infinitesimal generator of a C0-group Sε(t) on a Banach space X. If h:[t0,θ]×X→X and f·,·,·,ε:[t0,θ]×X×X→X are continuously differentiable on [t0,θ]×X and [t0,θ]×X×X, respectively, then the mild solution x∈C([t0,θ],X) of (9) with x0∈D(A^) is a classical solution of the semilinear ACP (9) with respect to control u.Proof.
Letx be the solution x∈C([t0,θ],X) of (9) with x0∈D(A^). Set(19)Ht,xt≡11+KDft,xt,ht,xt,ε-KI∫0sxτdτ+Xstt. The continuous differentiability of h and F implies the continuity differentiability of H. Define a function φ by(20)φt=Sεt-t0Ht0,xt0+A^Sεt-t0x0+∫t0tSεt-s∂∂sHs,xsds.It follows from the assumptions that φ∈C([t0,θ],X) and that the function α(t,x)≡∂/∂xHt,xx is continuous in t on [t0,θ] and uniformly Lipschitz in x and Corollary 6 assures that the integral equation(21)λt=φt+∫t0tSεt-sαs,λsdshas a unique solution λ∈C([t0,θ],X). Moreover, from our assumption, we obtain(22)Ht,xt+δ-Ht,xt=∂∂xHt,xtxt+δ-xt+ot,δ,Ht+δ,xt+δ-Ht,xt+δ=∂∂tHt,xt+δδ+ot,δ. This implies that(23)Ht+δ,xt+δ-Ht,xt=∂∂xHt,xtxt+δ-xt+∂∂tHt,xt+δδ+ot,δ,where o(t,·) denotes a little-o notation and δ-1o(t,δ)→0 as δ→0 uniformly on [0,θ]. We will show that x(t) is differentiable on [0,θ] by regarding the convergence of vδ(t)≡δ-1x(t+δ)-x(t)-λ(t) as δ→0 in norm ·. It follows from the definition of x, (20), (21), and (23), that (24)vδt=δ-1xt+δ-xt-λt=δ-1Sεt+δx0+∫t0t+δSεt+δ-sHs,xs-Sεtx0-∫t0tSεt-sHs,xs-Sεt-t0Ht0,xt0-A^Sεt-t0x0-∫t0tSεt-s∂∂sHs,xsds-∫t0tSεt-s∂∂xHt,xsλsds=δ-1Sεt-t0+δx0-Sεt-t0x0-A^Sεt-t0x0+δ-1∫t0-δt0Sεt-sHs+δ,xs+δ-Sεt-t0Ht0,xt0ds+δ-1∫t0tSεt-sHs+δ,xs+δ-Hs,xsds-∫t0tSεt-s∂∂sHs,xsds-∫t0tSεt-s∂∂xHt,xsλsds=δ-1Sεt-t0+δx0-Sεt-t0x0-A^Sεt-t0x0+δ-1∫t0-δt0Sεt-sHs+δ,xs+δ-SεtHt0,xt0ds+δ-1∫t0tSεt-s∂∂xHs,xsxs+δ-xs+∂∂sHs,xs+δδ+os,δds-∫t0tSεt-s∂∂sHs,xsds-∫t0tSεt-s∂∂xHt,xsλsds=δ-1Sεt-t0+δx0-Sεt-t0x0-A^Sεt-t0x0+δ-1∫t0-δt0Sεt-sHs+δ,xs+δ-Sεt-t0Ht0,xt0ds+δ-1∫t0tSεt-s∂∂xHs,xsxs+δ-xsds+δ-1∫t0tSεt-sos,δds-∫t0tSt-s∂∂xHt,xsλsds=δ-1Sεt-t0+δx0-Sεt-t0x0-A^Sεt-t0x0+δ-1∫t0-δt0Sεt-sHs+δ,xs+δ-Sεt-t0Ht0,xt0ds+∫0tSεt-s∂∂xHs,xsδ-1xs+δ-xs-λsds+δ-1∫0tSεt-sos,δds≤δ-1Sεt-t0+δx0-Sεt-t0x0-A^Sεt-t0x0+δ-1∫t0-δt0Sεt-sHs+δ,xs+δ-Sεt-t0Ht0,xt0ds+δ-1∫t0tSεt-sos,δds+∫t0tSεt-s∂∂xHs,xsvδsds. Since δ-1Sε(t-t0+δ)x0-Sε(t-t0)x0-A^Sε(t-t0)x0→0, δ-1∫t0tSε(t-s)o(s,δ)ds→0 and δ-1∫t0-δt0Sε(t-s)Hs+δ,x(s+δ)-Sε(t)Ht0,x(t0)ds→0 as δ→0, (25)δ-1Sεt-t0+δx0-Sεt-t0x0-A^Sεt-t0x0+δ-1∫t0tSεt-sos,δds+δ-1∫-δt0Sεt-sHs+δ,xs+δ-Sεt-t0Ht0,xt0ds≤βεδ for some nonnegative function βε(δ)→0 as δ→0. Therefore, we have(26)vδt≤βεδ+Mε∫t0tvδsds,where Mε=supt0≤s≤θ⁡Sε(t-s)∂/∂xHs,x(s). By applying Gronwall’s lemma to (26), we have(27)vδt≤βδexp⁡θMε. Thus vδ(t)=δ-1x(t+δ)-x(t)-λ(t)→0 as δ→0. This implies that x(t) is differentiable on [t0,θ] and its derivative is λ(t) which is an element in C([t0,θ],X). Consequently, x(t) is continuously differentiable on [t0,θ]. Next, we will show that x(t) satisfies the semilinear ACP (9). Since x(t) is differentiable on [t0,θ] and Sε(t) is C0-group,(28)x′t=ddtSεt-t0x0+11+KDddt∫t0tSεt-sfs,xs,hs,xs,ε-KI∫t0sxτdτ+Xspsds=A^Sεt-t0x0+11+KDddtSεt∫t0tSε-sfs,xs,hs,xs,ε-KI∫t0sxτdτ+Xspsds=A^Sεt-t0x0+11+KDSεtSε-tft,xt,ht,xt,ε-KI∫t0sxτdτ+Xspt+11+KDA^Sεt∫t0tS-sfs,xs,hs,xs-KI∫t0sxτdτ+Xspsds=A^Sεt-t0x0+11+KDSεt∫t0tSε-sfs,xs,hs,xs,ε-KI∫t0sxτdτ+Xspsds+11+KDft,xt,ht,xt,ε-KI∫t0txτdτ+Xspt=A^xt+11+KDft,xt,ht,xt,ε-KI∫t0txτdτ+Xspt.Hence x(t) satisfies the semilinear ACP (9). Finally, we show that x(t)∈D(A^) for all t∈[t0,θ]. It follows from (28) and x0∈D(A) that(29)A^xt=x′t-11+KDft,xt,ht,xt,ε-KI∫t0txτdτ+Xspt. Using the assumption of F and h, and the continuity of x′, we obtain the result that A^x(t) is continuous. This implies that x(t)∈D(A^) for all t∈[t0,θ].


### 3.3. System with Impulses

Next, we prove the existence of solution of the impulsive system (8). Throughout this section, we denote PC([0,T],X) as all type I-piecewise continuous function from [0,T] to X such with discontinuous at ti,i=1,…,n.Definition 9.
Letu∈Aad. A mild solution on [0,T] of the semilinear impulsive ACP (8) with respect to control u is a function x∈PC([0,T],X) such that x(t) satisfies the integral equation(30)xt=Sεtx0+11+KD∫0tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds+∑0<tk≤tSεt-tkIkxtk,htk,xtk, where Sε(t)t≥0 is the C0-semigroup generated by the operator A^=A(ε)-KPI/1+KD.Definition 10.
Letu∈Aad. A classical solution of the inhomogeneous ACP (8) with respect to control u is a function x∈PC([0,T],X) such that (1)
x is continuously differentiable on (0,t1), (tk-1,tk), and (tn,T) for all k=1,2,…,n;(2)
x(t)∈D(A^) for all t∈[0,T];(3)
x satisfies the semilinear ACP (8).Theorem 11.
Letu∈Aad. Let A^=A(ε)-KpI/1+KD be the infinitesimal generator of a C0-group Sε(t) on a Banach space X. If h:[t0,θ]×X→X is continuously differentiable on (0,t1)×X, (tk-1,tk)×X and (tn,T)×X for all k=1,2,…,n and f(·,·,·,ε):[t0,θ]×X×X→X is continuously differentiable on (0,t1)×X×X,  (tk-1,tk)×X×X, and (tn,T)×X×X for all k=1,2,…,n, then the mild solution x∈PC([0,T],X) of (8) with x0,Ik(x(tk),h(tk,x(tk)))∈D(A^) for all k=1,2,…,n, is a classical solution of the semilinear ACP (8) with respect to control u.Proof.
Let us consider the following system:(31)x˙t=A^xt+11+KDft,xt,ht,xt,ε-KI∫0txτdτ+Xspt,0≤t≤t1,x0=x0.Theorems 5 and 8 imply that System (31) has a unique mild solution x∈C([0,t1],X) such that(32)xt=Sεtx0+11+KD∫0tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds and the mild solution x(t) is a classical solution on [0,t1]. So(33)xt1=Sεt1x0+11+KD∫0t1Sεt1-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds. Next, we consider the following system:(34)x˙t=A^xt+11+KDft,xt,ht,xt,ε-KI∫0txτdτ+Xspt,t1≤t≤t2,xt1=Sεt1x0+11+KD∫0t1Sεt1-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds+I1xt1,ht1,xt1. By applying Theorems 5 and 8, we have(35)xt=Sεt-t1Sεt1x0+11+KD∫0t1Sεt1-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds+I1xt1,ht1,xt1+11+KD∫t1tSεt-sFs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds=Sεtx0+11+KD∫0t1Sεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds+Sεt-t1I1xt1,ht1,xt1+11+KD∫t1tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds=Sεtx0+11+KD∫0tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds+Sεt-t1I1xt1,ht1,xt1. That is,(36)xt=Sεtx0+11+KD∫0tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds+Sεt-t1I1xt1,ht1,xt1 is a classical solution on [t1,t2]. Continuing this process, we have that(37)xt=Sεtx0+11+KD∫0tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds+∑0<tk≤tSεt-tkIkxtk,htk,xtk is a classical solution on [0,T].


## 3.1. Approximation Method

Suppose that(x0,y0) is arbitrary initial value. Then we can choose an initial (x0,y0) such that the fast equation becomes stationary, that is, y˙(t;y0)=0. This implies that(5)Bεy+gt,x,y,ε=0. Accordingly, for small enough ε, it seems reasonable to substitute (3) with a differential algebraic equation:(6)x˙=Aεxt+ft,xt,yt,ε+ut,t≠tk,0=Byt+Gt,xt,yt,t≠tk,Δxtk=Ikxtk,ytk,Δytk=Jkxtk,ytk,x0=x0,y0=y0,where B=B(0) and G(t,x(t),y(t))=g(t,x(t),y(t),0).Suppose thaty(t)=h(t,x(t))∈D(B)⊆X solves the following system:(7)0=Byt+Gt,xt,yt,t≠tk,Δytk=Jkxtk,ytk,y0=y0. Therefore, by substituting y(t)=h(t,x(t)) into (6), we have(8)x˙=Aε-KPI1+KDxt+11+KDft,xt,ht,xt,ε-KI∫0txτdτ+Xspt,t≠tk,Δxtk=Ikxtk,htk,xtk,x0=x0,where Xsp(t)=KPxsp(t)+KI∫0txspτdτ+KDx′sp(t). In other words, we can approximate the slow equation by the inhomogeneous ACP (8).First, we solve semilinear ACP:(9)x˙=Aε-KPI1+KDxt+11+KDft,xt,ht,xt,ε-KI∫t0txτdτ+Xspt,t0≤t≤θ,xt0=x0.Let Aad=u∣u(t)=KPe(t)+KI∫0te(τ)dτ+KDde(t)/dt,KP,KI,KD≥0,KD≠-1 be an admissible control set.Definition 4.
Letu∈Aad. A mild solution on [t0,θ] of the semilinear ACP (9) with respect to control u is a continuous function x:[t0,τ]→X such that x(t) satisfies the integral equation:(10)xt=Sεt-t0x0+11+KD∫t0tSεt-sfs,xs,hs,xs,ε-KI∫t0sxτdτ+Xspsds, where Sε(t)t≥0 is the C0-semigroup generated by the operator A^=A(ε)-KP/1+KD.We start with the following classical result which assures the existence and uniqueness of mild solutions of (9) for Lipschitz continuous function h and F.Theorem 5.
Letu∈Aad. Let h:[t0,θ]×X→X,  f·,·,·,ε:[t0,θ]×X×X→X be continuous in t on [t0,θ] and uniformly Lipschitz continuous (with constant Lh and LF,ε) on X. If A^=A(ε)-KPI/1+KD is infinitesimal generator of a C0-semigroup Sε(t)t≥0 on X, then for every x0∈X the semilinear ACP (9) has a unique mild solution x∈C([t0,θ],X) with respect to control u. Moreover, the mapping x0→x is Lipchitz continuous from X into C([t0,θ],X).Proof.
For a givenx0∈X. Define a mapping Q:C([t0,θ],X)→C([t0,θ],X) by(11)Qxt=Sεt-t0x0+11+KD∫t0tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xstsds.Denoting by x∞ the supremum norm of x as an element of C([t0,θ],X), it follows from definition of Q and the uniformly Lipschitz continuous of h and f that(12)Qx1t-Qx2t≤Mε1+KD∫t0tLF,εx1s-x2s+hs,x1s-hs,x2s+KI∫t0sx1τ-x2τdτds≤Mε1+KDLF,ε1+Lh+KIt-t0t-t0x1-x2∞≤Mε1+KDLF,ε1+Lhθ+KIθ2x1-x2∞,where Mε is a bound of Sε(t) on [t0,θ]. Using (11) and (12) and induction on n, we have(13)Qnx1t-Qnx2t≤Mε/1+KDLF,ε1+Lhθ+KIθ2nn!x1-x2∞. This implies that(14)Qnx1-Qnx2∞≤Mε/1+KDLF,ε1+Lhθ+KIθ2nn!x1-x2∞. For sufficiently large n,  Mε/1+KDLF,ε1+Lhθ+KIθ2n/n!<1 and by a well-known extension to the contraction mapping principle, there is a unique x∈C([t0,θ],X) such that(15)xt=Qxt=Sεt-t0x0+11+KD∫t0tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xstsds. Therefore, the semilinear ACP (9) has a unique mild solution x∈C([t0,θ],X). The Lipchitz continuity of the mapping x0→x is consequences of the following argument. Let x1 and x2 be the mild solutions of (9) with the initial values x10 and x20, respectively. Then it follows from the definition of mild solution and the uniformly Lipschitz continuous of h and f that(16)x1t-x2t≤Mεx10-x20+LF,ε1+Lh+KIC1+KD∫t0tx1s-x2sds for some constant C>0. By using Gronwall’s lemma, it can be implied that(17)x1-x2∞≤Mεexp⁡LF,ε1+Lh+KIC1+KDθx10-x20 which yields the Lipchitz continuity of the mapping x0→x.Corollary 6.
Under the conditions of Theorem5, for every φ∈C[t0,θ],X, the integral equation(18)λt=φt+11+KD∫t0tSεt-sfs,λs,hs,λs,ε-KI∫0sλτdτ+Xstsds has a unique solution λ∈C[t0,θ],X.Proof.
The proof is similar to the proof of Theorem5 by defining (Qλ)(t)=φ(t)+1/1+KD∫t0tSε(t-s)f(s,λ(s),hs,λs,ε)-KI∫0sλ(τ)dτ+Xst(s)ds.


## 3.2. Regularity of the System without Impulses

Definition 7.
Letu∈Aad. A classical solution of the inhomogeneous ACP (9) with respect to control u is a function x∈C[t0,θ],X such that (1)
x is continuously differentiable on (t0,θ);(2)
x(t)∈D(A) for all t∈[t0,θ];(3)
x satisfies the semilinear ACP (9).Theorem 8 (regularity).
Letu∈Aad and A^=A(ε)-KpI/1+KD be the infinitesimal generator of a C0-group Sε(t) on a Banach space X. If h:[t0,θ]×X→X and f·,·,·,ε:[t0,θ]×X×X→X are continuously differentiable on [t0,θ]×X and [t0,θ]×X×X, respectively, then the mild solution x∈C([t0,θ],X) of (9) with x0∈D(A^) is a classical solution of the semilinear ACP (9) with respect to control u.Proof.
Letx be the solution x∈C([t0,θ],X) of (9) with x0∈D(A^). Set(19)Ht,xt≡11+KDft,xt,ht,xt,ε-KI∫0sxτdτ+Xstt. The continuous differentiability of h and F implies the continuity differentiability of H. Define a function φ by(20)φt=Sεt-t0Ht0,xt0+A^Sεt-t0x0+∫t0tSεt-s∂∂sHs,xsds.It follows from the assumptions that φ∈C([t0,θ],X) and that the function α(t,x)≡∂/∂xHt,xx is continuous in t on [t0,θ] and uniformly Lipschitz in x and Corollary 6 assures that the integral equation(21)λt=φt+∫t0tSεt-sαs,λsdshas a unique solution λ∈C([t0,θ],X). Moreover, from our assumption, we obtain(22)Ht,xt+δ-Ht,xt=∂∂xHt,xtxt+δ-xt+ot,δ,Ht+δ,xt+δ-Ht,xt+δ=∂∂tHt,xt+δδ+ot,δ. This implies that(23)Ht+δ,xt+δ-Ht,xt=∂∂xHt,xtxt+δ-xt+∂∂tHt,xt+δδ+ot,δ,where o(t,·) denotes a little-o notation and δ-1o(t,δ)→0 as δ→0 uniformly on [0,θ]. We will show that x(t) is differentiable on [0,θ] by regarding the convergence of vδ(t)≡δ-1x(t+δ)-x(t)-λ(t) as δ→0 in norm ·. It follows from the definition of x, (20), (21), and (23), that (24)vδt=δ-1xt+δ-xt-λt=δ-1Sεt+δx0+∫t0t+δSεt+δ-sHs,xs-Sεtx0-∫t0tSεt-sHs,xs-Sεt-t0Ht0,xt0-A^Sεt-t0x0-∫t0tSεt-s∂∂sHs,xsds-∫t0tSεt-s∂∂xHt,xsλsds=δ-1Sεt-t0+δx0-Sεt-t0x0-A^Sεt-t0x0+δ-1∫t0-δt0Sεt-sHs+δ,xs+δ-Sεt-t0Ht0,xt0ds+δ-1∫t0tSεt-sHs+δ,xs+δ-Hs,xsds-∫t0tSεt-s∂∂sHs,xsds-∫t0tSεt-s∂∂xHt,xsλsds=δ-1Sεt-t0+δx0-Sεt-t0x0-A^Sεt-t0x0+δ-1∫t0-δt0Sεt-sHs+δ,xs+δ-SεtHt0,xt0ds+δ-1∫t0tSεt-s∂∂xHs,xsxs+δ-xs+∂∂sHs,xs+δδ+os,δds-∫t0tSεt-s∂∂sHs,xsds-∫t0tSεt-s∂∂xHt,xsλsds=δ-1Sεt-t0+δx0-Sεt-t0x0-A^Sεt-t0x0+δ-1∫t0-δt0Sεt-sHs+δ,xs+δ-Sεt-t0Ht0,xt0ds+δ-1∫t0tSεt-s∂∂xHs,xsxs+δ-xsds+δ-1∫t0tSεt-sos,δds-∫t0tSt-s∂∂xHt,xsλsds=δ-1Sεt-t0+δx0-Sεt-t0x0-A^Sεt-t0x0+δ-1∫t0-δt0Sεt-sHs+δ,xs+δ-Sεt-t0Ht0,xt0ds+∫0tSεt-s∂∂xHs,xsδ-1xs+δ-xs-λsds+δ-1∫0tSεt-sos,δds≤δ-1Sεt-t0+δx0-Sεt-t0x0-A^Sεt-t0x0+δ-1∫t0-δt0Sεt-sHs+δ,xs+δ-Sεt-t0Ht0,xt0ds+δ-1∫t0tSεt-sos,δds+∫t0tSεt-s∂∂xHs,xsvδsds. Since δ-1Sε(t-t0+δ)x0-Sε(t-t0)x0-A^Sε(t-t0)x0→0, δ-1∫t0tSε(t-s)o(s,δ)ds→0 and δ-1∫t0-δt0Sε(t-s)Hs+δ,x(s+δ)-Sε(t)Ht0,x(t0)ds→0 as δ→0, (25)δ-1Sεt-t0+δx0-Sεt-t0x0-A^Sεt-t0x0+δ-1∫t0tSεt-sos,δds+δ-1∫-δt0Sεt-sHs+δ,xs+δ-Sεt-t0Ht0,xt0ds≤βεδ for some nonnegative function βε(δ)→0 as δ→0. Therefore, we have(26)vδt≤βεδ+Mε∫t0tvδsds,where Mε=supt0≤s≤θ⁡Sε(t-s)∂/∂xHs,x(s). By applying Gronwall’s lemma to (26), we have(27)vδt≤βδexp⁡θMε. Thus vδ(t)=δ-1x(t+δ)-x(t)-λ(t)→0 as δ→0. This implies that x(t) is differentiable on [t0,θ] and its derivative is λ(t) which is an element in C([t0,θ],X). Consequently, x(t) is continuously differentiable on [t0,θ]. Next, we will show that x(t) satisfies the semilinear ACP (9). Since x(t) is differentiable on [t0,θ] and Sε(t) is C0-group,(28)x′t=ddtSεt-t0x0+11+KDddt∫t0tSεt-sfs,xs,hs,xs,ε-KI∫t0sxτdτ+Xspsds=A^Sεt-t0x0+11+KDddtSεt∫t0tSε-sfs,xs,hs,xs,ε-KI∫t0sxτdτ+Xspsds=A^Sεt-t0x0+11+KDSεtSε-tft,xt,ht,xt,ε-KI∫t0sxτdτ+Xspt+11+KDA^Sεt∫t0tS-sfs,xs,hs,xs-KI∫t0sxτdτ+Xspsds=A^Sεt-t0x0+11+KDSεt∫t0tSε-sfs,xs,hs,xs,ε-KI∫t0sxτdτ+Xspsds+11+KDft,xt,ht,xt,ε-KI∫t0txτdτ+Xspt=A^xt+11+KDft,xt,ht,xt,ε-KI∫t0txτdτ+Xspt.Hence x(t) satisfies the semilinear ACP (9). Finally, we show that x(t)∈D(A^) for all t∈[t0,θ]. It follows from (28) and x0∈D(A) that(29)A^xt=x′t-11+KDft,xt,ht,xt,ε-KI∫t0txτdτ+Xspt. Using the assumption of F and h, and the continuity of x′, we obtain the result that A^x(t) is continuous. This implies that x(t)∈D(A^) for all t∈[t0,θ].


## 3.3. System with Impulses

Next, we prove the existence of solution of the impulsive system (8). Throughout this section, we denote PC([0,T],X) as all type I-piecewise continuous function from [0,T] to X such with discontinuous at ti,i=1,…,n.Definition 9.
Letu∈Aad. A mild solution on [0,T] of the semilinear impulsive ACP (8) with respect to control u is a function x∈PC([0,T],X) such that x(t) satisfies the integral equation(30)xt=Sεtx0+11+KD∫0tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds+∑0<tk≤tSεt-tkIkxtk,htk,xtk, where Sε(t)t≥0 is the C0-semigroup generated by the operator A^=A(ε)-KPI/1+KD.Definition 10.
Letu∈Aad. A classical solution of the inhomogeneous ACP (8) with respect to control u is a function x∈PC([0,T],X) such that (1)
x is continuously differentiable on (0,t1), (tk-1,tk), and (tn,T) for all k=1,2,…,n;(2)
x(t)∈D(A^) for all t∈[0,T];(3)
x satisfies the semilinear ACP (8).Theorem 11.
Letu∈Aad. Let A^=A(ε)-KpI/1+KD be the infinitesimal generator of a C0-group Sε(t) on a Banach space X. If h:[t0,θ]×X→X is continuously differentiable on (0,t1)×X, (tk-1,tk)×X and (tn,T)×X for all k=1,2,…,n and f(·,·,·,ε):[t0,θ]×X×X→X is continuously differentiable on (0,t1)×X×X,  (tk-1,tk)×X×X, and (tn,T)×X×X for all k=1,2,…,n, then the mild solution x∈PC([0,T],X) of (8) with x0,Ik(x(tk),h(tk,x(tk)))∈D(A^) for all k=1,2,…,n, is a classical solution of the semilinear ACP (8) with respect to control u.Proof.
Let us consider the following system:(31)x˙t=A^xt+11+KDft,xt,ht,xt,ε-KI∫0txτdτ+Xspt,0≤t≤t1,x0=x0.Theorems 5 and 8 imply that System (31) has a unique mild solution x∈C([0,t1],X) such that(32)xt=Sεtx0+11+KD∫0tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds and the mild solution x(t) is a classical solution on [0,t1]. So(33)xt1=Sεt1x0+11+KD∫0t1Sεt1-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds. Next, we consider the following system:(34)x˙t=A^xt+11+KDft,xt,ht,xt,ε-KI∫0txτdτ+Xspt,t1≤t≤t2,xt1=Sεt1x0+11+KD∫0t1Sεt1-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds+I1xt1,ht1,xt1. By applying Theorems 5 and 8, we have(35)xt=Sεt-t1Sεt1x0+11+KD∫0t1Sεt1-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds+I1xt1,ht1,xt1+11+KD∫t1tSεt-sFs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds=Sεtx0+11+KD∫0t1Sεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds+Sεt-t1I1xt1,ht1,xt1+11+KD∫t1tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds=Sεtx0+11+KD∫0tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds+Sεt-t1I1xt1,ht1,xt1. That is,(36)xt=Sεtx0+11+KD∫0tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds+Sεt-t1I1xt1,ht1,xt1 is a classical solution on [t1,t2]. Continuing this process, we have that(37)xt=Sεtx0+11+KD∫0tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds+∑0<tk≤tSεt-tkIkxtk,htk,xtk is a classical solution on [0,T].


## 4. Optimal Control Problem

In this section, we study an optimal control problem of the approximated system (8) such that is controlled by a controller variable u∈Aad. We denote each solution x with respect to a control u by xu and call the ordered pair (u,xu) a pairwise control pair, sometimes written shortly as (u,x). Define a cost functional J by integrating an error function, that is,(38)Ju,xu=∫0Texutdt, where e(xu(t)) is the error function. We want to investigate an optimization control problem, Problem (P).Problem (P). Problem (P) is to find the pairwise control pair (u^,x^)∈Aad×PC[0,T],X such that J(u^,x^)≤J(u,xu) for all u∈Aad.Theorem 12.
Suppose that the error functione(·) satisfies the following conditions: (1)
e(·):PC([0,T],X)→(-∞,∞] is Borel measurable.(2)
e(·) is sequentially lower semicontinuous on PC([0,T],X).(3)
e(·) is convex on PC([0,T],X).(4)
There are constantsa,b>0 such that e(x(t))≥a|x(t)|+b|u(t)| for all x∈PC([0,T],X) and u∈Aad.
Under the assumptions of Theorem11, Problem (P) constrained by system (8) has at least one solution; that is, there exists a pairwise control pair (u0,x0)∈Aad×PC[0,T],X such that J(u^,x^)≤J(u,xu) for all (u,xu)∈Aad×PC[0,T],X.Proof.
Letm=inf⁡J(u,xu)∣u∈Aad. If m=+∞, the theorem is already true. Assume that m<+∞. By condition (4), there are constants a,b>0 such that e(x(t))≥a|x(t)|+b|u(t)| for all x∈PC([0,T],X) and u∈Aad. Then (39)Ju,xu=∫0Texutdt≥a∫0Txutdt+b∫0Tutdt≥-m1>-∞, for some m1>0 and for all u∈Aad. The minimum of J implies that there is a sequence of the minimum point, say un, of the cost functional J such that limn→∞⁡J(xun,un)=m and J(un,xun)≥a∫0T|xun(t)|dt+b∫0T|un(t)|dt. Thus, there is N>0 and m2>0 such that m+m2≥J(xun,un)≥b∫0T|un(t)|dt for all n≥N. This implies that m+m2/b≥∫0T|un(t)|dt. Consequently, un≤m+m2/Tb for all n≥N0. Hence un is a bounded sequence in Aad⊂L2[0,T],X with respect to the norm ·. Since L2[0,T],X is a reflexive Banach space and Aad is unit ball of L2[0,T],X, there exists a subsequence unk of un such that unk→wu^ for some u^∈Aad. Let x^∈PC([0,T],X) be a mild solution with respect to a control u^ and let xnk be a sequence of mild solution corresponding to the sequence of control unk, that is,(40)xnkt=Sεtx0+11+KD∫0tSεt-sfs,xnks,hs,xnks,ε-KI∫0sxnkτdτ+Xspsds+∑0<tj≤tSεt-tjIjxnktj,htj,xnktj. By the continuous differentiability of f and h, there is a constant L>0 such that (41)xnk-x^≤L∫0txnks-x^sds+unk-u^. By Gronwall’s lemma, there is an L1>0 such that (42)xnk-x^≤L1unk-u^. Since unk→wu0,  xnk→wx0. By using conditions (2) and (3), (43)m=limnk→∞¯⁡Jxnkunk,unk=limnk→∞¯⁡∫0Texnktdt≥∫0Telimnk→∞¯⁡xnktdt=Ju^,x^. Thus, J(u^,x^)=m, that is, J(u^,x^)≤J(u,xu) for all (u,xu)∈Aad×PC[0,T],X.


## 5. Example

In this section, an example is presented to clarify the main results. Consider a perturbed differential system with control inR2, for all t∈[0,3]:(44)dx1dt=-εx1+x2+x3+u1,∀t≠1,2,dx2dt=x1+x2-2εx2t+u2,∀t≠1,2,dx3dt=1εx3-x1+ε,∀t≠1,2,Δx1n=x1n2,Δx2n=x2+yn2,Δx3n=x1n2,x10=0,x20=1,x30=0,where u(t)=u1(t)u2(t)=KPe(t)+KI∫0te(τ)dτ+KDde(t)/dt and e(t)=x1(t)x2(t). Assume that a set point function of the example is xsp=00. Let x=x1,x2 and y=x3. Then system (44) becomes(45)x˙1x˙2=-ε111x1x2+y-2εx2t+u1u2,εy˙=y-x1+ε,Δxn=1n2x1x2+y,Δyn=x1n2,x0=01,y0=0.Since y=x1 is a solution of εy˙=y-x1+ε with Δy(n)=x1/n2 as ε→0, we can approximate (45) with(46)x˙1x˙2=-ε111x1x2+x1-2εx2t+u1u2,Δxn=1n2x1nx2n+x1n,x0=01,where u1u2=KPx1x2+KI∫0tx1(τ)x2(τ)dτ+KDx˙1x˙2. Then, (46) satisfy the condition of Theorem 11. Therefore, an approximate mild solution of system (45) is in the following form:(47)xt=Sεt01+11+KD∫0tSεt-sx1s-2εsx2s-KI∫0sx1τx2τdτds+∑0<n≤tSεt-n1n2x1nx2n+x1n,where Sε(t)=exp⁡Aε-KPI/1+KDt=exp⁡t/1+KD-ε-KP111-KP. One can apply Picard iteration to (47). For example, we set KP=KD=KI=1. Then the iteration of approximation mild solution is (48)x1,ntx2,nt=Sεt01+12∫0tSεt-sx1,n-1s-2εsx2,n-1s-∫0sx1,n-1τx2,n-1τdτds+∑0<k≤tSεt-k1k2x1,n-1kx2,n-1k+x1,n-1k, where Sε(t)=exp⁡t/2-ε-1110=aCA-DB-2aA-B-2aA-B-aDA-CB with a=2ε2+2ε+5-1,  A=e-tC/4 and B=e-tD/4,  C=1+a+ε,  D=ε-a+1.If we set an initial functionx1,0(t)x2,0(t)=01 and ε=0.01, then, for 0<t≤3,(49)x1,1tx2,1t=3.487532+1.948632e-0.308290t-5.436164e-0.196710t-0.418806t2.684795+0.766632e-0.308290t-3.577218e-0.196710t-0.422994t+∑0<k≤t1k2-0.446318e-0.308290t-k-e-0.196710t-k-0.175591e-0.308290t-k+0.275191e-0.196710t-k. We can continue the process until this condition is met “if Xn-Xn-1<δ1 then Xn-xsp<δ2, for some small given δ1,  δ2 where Xn=x1,nx2,n”. In fact, the value of Xn-xsp is called deviation concerning with manipulated variable (the manipulated variable is the variable that causes a change in the controlled variable) that we want to eliminate. Define a cost functional J by(50)Ju,xu=∫0Txut-xsptdt.Problem (P) is to find the pairwise control pair (u^,x^)∈Aad×PC[0,T],X such that J(u^,x^)≤J(u,xu) for all u∈Aad. We use Theorem 12 to confirm that Problem (P) constrained by system (45) has at least one solution.


## 6. Conclusion

In this study, we consider the PID control system of the impulsive differential equation with singular perturbation. Because of the complexity of the system, it is difficult to find its exact solution. So the approximation method is introduced for solving the system. The idea of the method is to eliminate the fast variable in the system under some assumptions. We found an approximated mild solution of the system from the approximation method. The approximated mild solution of the system is written in the following form:(51)xt=Sεtx0+11+KD∫0tSεt-sfs,xs,hs,xs,ε-KI∫0sxτdτ+Xspsds+∑0<tk≤tSεt-tkIkxtk,htk,xtk, where A^=A(ε)-KP/1+KD is the infinitesimal generator of a C0-semigroup {Sε(t)}t≥0. We showed that the mild solution is also the classical solution of the approximated system. Moreover, we proved the theorem confirming that Problem (P) constrained by the PID control system has at least one solution. In this article, we do not mention how to design controllers, but it is well known that there are many different ways to design PID control (please see [21–23, 25] and references therein). The result form this paper can be used as a prerequisite to PID controller design. For example, the C0-semigroup property of the operator A^ will be a necessary condition for evaluating the controller parameters KI, KP, and KD that play important role in PID controller design process.



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*Source: 1938513-2017-11-12.xml*