Optimal control of noninstantaneous impulsive differential equations

Accepted Manuscript

Optimal Control of Noninstantaneous Impulsive Differential Equations Shengda Liu, JinRong Wang, Yong Zhou PII: DOI: Reference:

S0016-0032(17)30479-9 10.1016/j.jfranklin.2017.09.010 FI 3146

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

16 November 2016 30 August 2017 17 September 2017

Please cite this article as: Shengda Liu, JinRong Wang, Yong Zhou, Optimal Control of Noninstantaneous Impulsive Differential Equations, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.09.010

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Optimal Control of Noninstantaneous Impulsive Differential Equations Shengda Liua , JinRong Wanga,∗, Yong Zhoub,c,∗ a Department

of Mathematics, Guizhou University, Guiyang, Guizhou 550025, P.R. China of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, P.R. China Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

c

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b Department

Abstract

In this paper, we study optimal control problems for a new class of noninstantaneous impulsive differential equations arising from the dynamics of evolution processes in pharmacotherapy. We construct

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a suitable control function, which allows us to characterize the structure of controllability by using the terminal time subinterval instead of the global time interval. We apply fixed point approach to show the controllability results that are the foundation of optimal control theory. Next, we study existence of optimal control problems for a certain quadratic functional acting as the performance index. In addition, we design ILC updating laws for deterministic impulsive systems to generate a sequence of control functions to approximate the optimal control function. Further, we extend the

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deterministic results to random case by designing ILC updating laws with randomly varying trial length. Finally, several examples are given to demonstrate the validity of theoretical results and

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design methods. Here, we remark that ILC updating algorithm is adopted to find a desired optimal control for impulsive systems, which provides another effective way to solve optimization problem via computer techniques.

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Keywords: Noninstantaneous impulsive differential equations, Controllability, Optimal control,

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Iterative learning control, Randomly varying trial length.

1. Introduction

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It is well known that the classical instantaneous impulsive differential equations (differential equations with fixed impulsive moments) have gained considerable popularity and importance in the last century. In fact, the class of instantaneous impulsive differential equations of various types are widely used in dynamical systems, control system and physics to formulate many mathematical

✩ The first and second author’s work was partially supported by National Natural Science Foundation of China (11661016). The third author acknowledges the support by National Natural Science Foundation of China (11271309,11671339). ∗ Corresponding author. Email addresses: [email protected] (Shengda Liu), [email protected] (JinRong Wang), [email protected] (Yong Zhou)

Preprint submitted to Journal of the Franklin Institute

September 26, 2017

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modelings. Concerning on the theoretical and application of instantaneous impulsive differential equations, we refer to D. D. Bainov and P. S. Simeonov [1, 2, 3], V. Lakshmikantham et al. [4], A. M. Samoilenko et al. [5], M. Benchohra et al. [6] and reference therein. Note that the stability of impulsive systems simultaneously is one of the most interesting issues. In general, there are two different impulses, one is stabilizing impulse and the other one is destabilizing impulse (see [7, 8, 9]). Next, the evolutionary process of many mathematical models whose motions depend on some

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abrupt changes in their states is best described by differential equations with noninstantaneous impulsive moments (noninstantaneous impulsive differential equations). However, the development of fundamental theory for noninstantaneous impulsive differential equations is still very slow. In 2013, D. O’Regan et al. [10, 11] studied the Cauchy problem for a new type semilinear evolution equations with noninstantaneous impulsive moments, where impulsive moments start at the fixed impulsive point ti and continue to act on a finite time interval [ti , si ]. About the qualitative theory

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of this new kind of noninstantaneous impulsive differential equations, especially on the existence of solutions via fixed point approach, periodic solutions and stability via Poinc´ are operator, we would like to recommend the reader to see [12, 13, 14, 15, 16, 17, 18, 19, 20]. We also remark that controllability and optimal control results for such new class of noninstantaneous impulsive differential equations are still not been reported fully (see [21, 22, 23, 24]). The concept of controllability was proposed by R. E. Kalman [25] in 1963. It plays a mostly

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role in the field of mathematical control theory. In order to determine the controllability of linear systems, one can turn into consider a controllability (Grammian) matrix [26] and a PBH-test [27].

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Similar methods can be also used to deal with linear impulsive differential systems (see [28, 29, 30, 31]). For nonlinear systems, controllability results can be shown via contraction mapping principle, Schauder’s fixed point theorem and Schaefer’s fixed point theorem by constructing a suitable control

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function (see [32, 33, 34, 35]). In particular, it is summarized that the sufficient conditions for approximate controllability of various types of dynamic systems using Schauders fixed-point theorem Of course, Krasnoselskii’s fixed point theorem and Nussbaum’s fixed point

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by A. Babiarz [36].

theorem can also be used to investigate the controllability of nonlinear systems (see [37, 38, 39]). Once a system is controllable, it is natural to offer an interesting question: How to obtain a

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control function of achieving greater, faster, better and more economic results? While people want to choose the best way to complete the task in a certain sense, optimal control problem is proposed. The classical optimal control theory can be found in the monograph [26] of X. Li and J. Yong. Further, for stochastic system case, one can find some related results in the monograph [40] of J. Yong and X. Zhou. We also note that there is a rapid development in the existence of optimal controls of fractional order systems (see [41, 42]). Meanwhile, we would like to mention that B. S. Mordukhovich et al. (see [43, 45, 44]) study optimal control problems for systems governed by differential inclusions problems. Once we obtain the existence result of optimal controls, it is

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necessary to find an effective algorithm to obtain this optimal control function. We note that some numerical optimization methods are given in [46], which are based on the numerical solutions of differential equations. In addition, D. H. Owens [47] proposed a norm optimal iterative learning control technique with the help of optimization theory and present an effective to seeking optimal parameters in ILC updating laws. Note that the hemodynamic equilibrium of a human always in the condition of varying time and

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the injection of drugs in the bloodstream and their consequent absorption in the body are gradual, and continuous processes. Thus, this evolution process should be regarded as continuous impulsive action, i.e., starting at an arbitrary fixed point and persisting active over the global time interval. This is the main motivation to study nonautonomous evolution equations with noninstantaneous impulsive moments.

In this paper, we study controllability, optimal controls and ILC for a new class of noninstanta-

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neous impulsive differential equations arising from the dynamics of evolution processes in pharmacotherapy. We give controllability, optimal controls and ILC results and show their relationship. It is remarkable that ILC updating law is marked as an algorithm to find the desired optimal control. This provides another way to solve optimization problem. Next, we give a sketch of our work as follows.

Firstly, we develop the classical idea [32] to transfer controllability problem into a fixed point

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problem for a new constructed operator equation. The constructed method is standard, however, the technique is nontrivial. Two sufficient conditions are established to guarantee our problem

mathematical analysis.

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is controllable via contraction mapping principle and Schauder’s fixed point theorem with rigor

Secondly, we develop the method in [24, 26] to study existence of optimal control problems for

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a certain quadratic functional acting as the performance index. Moreover, solving optimal control problems have attracted by many researchers [48, 49, 50]. In order to find an optimal control function

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for a class of trajectory tracking problems, we adopt the ILC updating technique that was proposed by M. Uchiyama [51] and S. Arimoto [52]. It has been widely studied in the past decades (see [53, 54, 55, 56, 57, 58, 59]). Here, we design ILC laws for determine noninstantaneous impulsive

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differential equations to construct a sequence of control functions to approximate the optimal control function. We are aim to provide an effective algorithm to make a cost of functional achieving its minimum via computer techniques. Thirdly, it does exit the case of operation terminated early, see an example in functional electrical

stimulation for upper limb movement, which strongly motivates us to consider ILC under randomly iteration-varying lengths environments (see [60, 61, 62, 63]). That is, the trial ends at non-uniform time of duration. Here, we present sufficient conditions for deterministic and random cases to guarantee the convergence of tracking error in the sense of a certain norm and the expectation of

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a certain norm of stochastic variable, respectively. We apply impulsive Gronwall’s inequality to complete the proof of convergence theorems. The main contributions of this paper are two folds. In comparison of the above related literatures, we use a new control function (see (2)) to show the structure of controllability involving the terminal time subinterval instead of the global time interval (see (3)). Although the method is standard in some sense, the constructed approach is technical

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(see (8)) for our problem based on a certain operator of Poincar´e type, a composition of the maps (see Lemma 2.5), and the techniques of a priori estimate (see Remarks 2.6, 2.7, 2.8). Two new controllability results (see Theorems 2.4, 2.12) are shown by virtue of nonlinear functional analysis and establishing Lemmas 2.9, 2.10, 2.11. Although the proof idea of existence of optimal controls (see Theorem 3.1) is adopted from [24], but now it is much more complicated for our problems since we need to study many time varying subsystems.

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We propose the ILC updating algorithm to generate a sequence of control functions, which are used to seek the desired optimal control function. We develop the design idea in [62, 63] to study ILC problem for systems with deterministic noninstantaneous impulses, then extend to offer ILC problem for a new class of systems varying with time and random noninstantaneous impulses via random trial length. With the help of the representation of solutions of involving evolution system, we apply open-loop P -type updating law with initial state learning and modify the classical definition

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of error to generate a sequences of control functions such that each output can track the given output trajectory in the standard λ-norm (see Theorem 4.1 and Corollary 4.2) and the expectation

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of λ-norm (see Theorem 5.1 and Corollary 5.2), respectively. We also use some explicit examples to demonstrate the validity of theoretical results and designed algorithm methods. With the help of the representation of solutions involving evolution matrix, sufficient conditions on learning gain

inequality.

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matrices are derived to guarantee the error tend to zero in the above sense via an impulsive Gronwall

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The rest of this paper is organized as follows. In Section 2, we give the sufficient condition of the controllability by contraction mapping principle and Schauder’s fixed point theorem. In Section 3, we give the existence of optimal controls. In Sections 4-5, we give the convergence analysis of

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iterative learning control in deterministic and random cases. Examples are given in Section 6 to demonstrate the application of our main results.

2. Controllability results Pn1 2
21 For a n1 -dimensional vector w = (w1 , w2 , · · · , wn1 )> , we define a vector norm kwk = . i=1 wi p The n1 × n1 matrix norm of A is defined as kAk = λmax (A> A), where λmax (·) is the maximum eigenvalue of ·. The C-norm and λ-norm of the function f : [0, T ] → Rn1 are defined as kf kC = supt∈[0,T ] kf (t)k and kf kλ = supt∈[0,T ] {kf (t)ke−λt } respectively. 4

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Let C([0, T ], Rn1 ) be the set of all continuous functions from [0, T ] into Rn1 . We introduce the piecewise continuous functions space P C([0, T ], Rn1 ) := {x : [0, T ] → Rn1 | x ∈ C((ti , ti+1 ], Rn1 ), i = + − 0, 1, · · · , N and ∃ x(t− i ) and x(ti ), i = 1, · · · , N, with x(ti ) = x(ti )} endowed with norm kf kP C =

supt∈[0,T ] kf (t)k. Obviously, (P C([0, T ], Rn1 ), k · kP C ) is a Banach space. Denote L (X, Y ) by the space of bounded linear operators from Banach space X to Banach space Y . Consider the following noninstantaneous impulsive controlled systems

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 SN   x(t) ˙ = A(t)x(t) + f (t, x(t)) + B(t)u(t), t ∈ j=0 [si , ti+1 ],      x(t) = B (t)x(t− ), t ∈ (t , s ], i = 1, 2, · · · , N, i i i i  x(s+ ) = x(s− ), i = 1, 2, · · · , N,   i i     x(0) = x , 0

(1)

C([0, T ] × Rn1 , Rn1 ), x(t), y(t), u(t) ∈ Rn1 .

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where 0 = s0 < t1 < s1 < · · · < sN −1 < tN < sN < tN +1 = T , A(·), B(·), Bi (·) ∈ C([0, T ], Rn1 ×n1 ), f ∈ Following [17, Definition 1], we give the following definition of solutions.

Definition 2.1. A function x ∈ P C([0, T ], Rn1 ) is called a solution of the problem (1) if x satisfies initial condition x(0) = x0 and

= U (t, si )x(si ) +

Z

t

U (t, s)[f (s, x(s)) + B(s)u(s)]ds,

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x(t)

si

x(s+ i )

= Bi (t)x(t− i ),

t ∈ (ti , si ],

i = 1, 2, · · · , N,

N [

[si , ti+1 ],

j=0

= x(s− i ), i = 1, · · · , N,

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x(t)

t∈

PT

where U (·, ·) denotes Cauchy matrix of x(t) ˙ = A(t)x(t), t ∈ [0, T ]. Definition 2.2. The system (1) is said to be controllable on [0, T ] if for every x0 , x1 ∈ Rn1 , there x1 .

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exists a control u ∈ P C([0, T ], Rn1 ) such that the solution x of (1) satisfying x(0) = x0 and x(T ) =

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We impose the following assumptions. [H1] The function f ∈ C([0, T ] × Rn1 , Rn1 ) and there exists a Lf > 0 such that kf (t, x) − f (t, y)k ≤ Lf kx − yk, t ∈ [0, T ], x, y ∈ Rn1 .

[U1] The linear equation x(t) ˙ = A(t)x(t) is well posed, i.e., the Cauchy matrix U (·, ·) ∈ L ({(t, s) : 0 ≤ s ≤ t ≤ T }, Rn1 ×n1 ). Denote MU = sup0≤s≤t≤T kU (t, s)k.

[B1] A(·), B(·), Bi (·) ∈ C([0, T ], Rn1 ×n1 ), i = 1, 2, · · · , N . 5

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Define Wi : L2 ([si , ti+1 ], Rn1 ) → Rn1 , i = 1, 2, · · · as follows: Wi (u) :=

 R  ti+1 U (t si

i+1 , s)B(s)u(s)ds,

 0,

Dom(u) ∩ [si , ti+1 ] 6= ∅,

Dom(u) ∩ [si , ti+1 ] = ∅,

where Dom(u) is the domain of u.

W :=

N −1 X

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Then we can get the operator W : L2 ([0, T ], Rn1 ) → Rn1 as follows: Wi χ[si ,ti ] .

i=0

where χ is the characteristic function.

Clearly, the linear operator Wi from L2 ([si , ti+1 ], Rn1 ) into Im(Wi ) ⊆ Rn1 . For the fundamental

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˜ i : u + ker(Wi ) 7−→ homomorphism theorem, we have L2 ([si , ti+1 ], Rn1 )/ ker(Wi ) ∼ = Im(Wi ) with W Wi (u). The symbol ker(W ) and Im(W ) are the kernels and images of the mapping W , respectively. Therefore, we assume that:

[C1] Wi : L2 ([si , ti+1 ], Rn1 ) → Rn has a bounded inverse operator Wi−1 and take values from L2 ([si , ti+1 ], Rn1 )/ ker(Wi ) and there exists a Lw > 0 such that kWi−1 [u]k ≤ Lw kuk, for all

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i = 0, 1, · · · , N − 1.

Clearly, Wi must be surjective to satisfy [C1]. On the other hand, if Wi is surjective then we can

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define an inverse Wi−1 : Rn → L2 ([si , ti+1 ], Rn )/ ker Wi . PN −1 Define an operator W −1 := i=0 Wi−1 χ[si ,ti ] . Following the similar methods in [12, Theorem

2.1] and [17, Theroem 4], we have the following existence and uniqueness result. Since the proof is

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standard, we omit it here.

Theorem 2.3. Assume [H1], [U1] and [B1] are satisfied. For any u ∈ L2 ([0, T ], Rn1 ), system (1)

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has a unique solution x ∈ P C([0, T ], R).

We adopt the standard framework to deal with controllability problems as follows. For an arbitrary function x and the point si , i = 1, 2, · · · , N , we define the following control

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function:

 Z usi ,x := W −1 x(ti+1 ) − U (ti+1 , si )B(si )x(t− ) − i

ti+1

si

 U (ti+1 , s)f (s, x(s))ds .

(2)

Define Pi,C : P C([0, T ], Rn1 ) → P C([0, T ], Rn1 ), i = 1, 2, · · · , N as follows: Pi,C [x(·)]

:= U (·, si )Bi (si )x(t− i )+

Z

·

U (·, s)[f (s, x(s)) + B(s)usi ,x (s)]ds.

(3)

si

In what follows, it is necessary to show that, when using the control usN ,x in (2), the operator PN,C 6

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defined in (3) has a fixed point. Obviously, one can check PN,C [x(T )] = U (T, sN )BN (sN )x(t− N) +

Z

T

U (T, s)[f (s, x(s)) + B(s)usN ,x (s)]ds

sN

sN T

−

Z

sN

 U (T, τ )f (τ, x(τ ))dτ ](s) ds = x1 .

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= U (T, sN )BN (sN )x(t− N)  Z T + U (T, s) f (s, x(s)) + B(s)W −1 [x(T ) − U (T, sN )B(sN )x(t− N)

Now we are ready to give the first controllability result by using contraction mapping principle. Theorem 2.4. Assume [H1], [U1], [B1] and [C1] are satisfied. If

then system (1) is controllable on [0, T ].

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  Υ := MU kBN (sN )k + MU Lf + kBkC Lw (1 + MU + MU Lf (T − sN )) (T − sN ) < 1,

Proof. For any x, x e ∈ P C([0, T ], Rn1 ), we have

kU (t, sN )kkBN (sN )kkx − x ekP C +  +kB(s)kkusN ,x − usN ,ex k ds.

Z

t

sN

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≤

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kPN,C [x(t)] − PN,C [e x(t)]k

 kU (t, s)k Lf kx(s) − x e(s)k

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Using assumption [H1], [B1] and [C1], one can obtain kusN ,x − usN ,ex k

AC

CE

≤ Lw kx(T ) − U (T, sN )BN (sN )x(t− N) Z T − U (T, s)f (s, x(s))ds sN

Z T −e x(T ) + U (T, sN )BN (sN )e x(t− ) + U (T, s)f (s, x e(s))dsk N sN  ≤ Lw kx(T ) − x e(T )k + kU (T, sN )kkBN (sN )kkx(t− e(t− N) − x N )k +

Z

T

s

≤

 kU (T, s)kkf (s, x(s)) − f (s, x e(s))kds

N Lw kx − x ekP C + kU (T, sN )kkBN (sN )kkx − x ekP C

+

Z

T

sN

 kU (T, s)kLf kx(s) − x e(s)kds 7

(4)

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 Z ≤ Lw 1 + kU (T, sN )kkBN (sN )k +

T

sN

Submitting (5) into (4), we have

 kU (T, s)kLf ds kx − x ekP C .

(5)

kPN,C [x(t)] − PN,C [e x(t)]k

sN

T

sN

which implies that

 kU (T, τ )kLf dτ ) dskx − x ekP C ,

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≤ kU (t, sN )kkBN (sN )kkx − x ekP C  Z t Z + kU (t, s)k Lf + kB(s)kLw (1 + kU (T, sN )k +

kPN,C [x] − PN,C [e x]kP C ≤ Υkx − x ekP C .

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Since Υ < 1, one can apply contraction mapping to derive our result. The proof is completed. Next, we present the second theorem of controllability via Schauder’s fixed point theorem. We assume the following hypotheses.

[C2] For each t ∈ [0, T ], the function f (t, ·) : Rn1 → Rn1 is continuous and the function f (·, x) :

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[0, T ] → Rn1 is strongly measurable for each x ∈ Rn1 .

[C3] There exists a constant Mf > 0 such that kf (t, x)k ≤ Mf (1 + kxk), t ∈ [0, T ], x ∈ Rn1 . max

i=1,2,··· ,N +1

ρi < 1,

max

i=1,2,··· ,N

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[C4] Suppose

where

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K ≥ max

CE

ρ1

AC

M1∗



kBi kC ≤ 1 and

 ∗ MN M1∗ M2∗ +1 , , ··· , , kx0 k, kx1 k , 1 − ρ1 1 − ρ2 1 − ρN +1

= MU [Mf + kBkC Lw (1 + MU Mf t1 )]t1 , = MU [kx0 k + Mf t1 + kBkC Lw (MU kx0 k + MU Mf t1 )t1 ],

and

ρi+1

=

MU (kBi kC + Mf (ti+1 − si ) +kBkC Lw (1 + MU kBi kC + MU Mf (ti+1 − si ))(ti+1 − si )),

∗ Mi+1

= MU (Mf (ti+1 − si ) + kBkC Lw MU Mf (ti+1 − si )2 ).

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Following the method in [17, (6)-(11)], for any z ∈ Rn1 , we define P0 (z)

=

x(t1 , z), x(t, z) = U (t, 0)z +

Z

t

U (t, s)[f (s, x(s, z)) + B(s)us0 ,x (s)]ds,

0

Gi (z)

=

Bi (si )z,

Pi (z)

=

x(ti+1 , z), x(t, z) = U (t, si )z +

Z

t

si

U (t, s)[f (s, x(s, z)) + B(s)usi ,x (s)]ds, i = 1, · · · , N.

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Lemma 2.5. For any z ∈ Rn1 , we have

kGj ◦ Pj−1 ◦ Gj−1 ◦ Pj−2 ◦ · · · ◦ G1 ◦ P0 (z)k ≤ a(j)(1 + b(j) + · · · + b(j)j−1 ) + b(j)j kzk,

=

b(j)

=

kukL2

=

max

i=1,2,··· ,j

max

i=1,2,··· ,j

Z

T

k

0

kBi kC MU eMU Mf tj+1 ,  kBi kC MU Mf max

i=1,2,··· ,N

N X

2

√ (ti+1 − si ) + kBkC T kukL2

! 21

usi ,x (t)χ[si ,ti+1 ] (t)k dt

i=0



eMU Mf tj+1 ,

.

M

a(j)

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where

(6)

for any j = 1, 2, · · · , N .

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Proof. For any t ∈ [si , ti+1 ], by (2), we derive that   Z t Lw kx(t)k + kU (t, si )Bi (si )x(t− )k + kf (s, x(s))kds i

≤

si

PT

kusi ,x (t)k

By [B1], [C1] and [C3], we have

CE

kusi ,x (t)k

AC

For any t ∈

≤

SN

i=0 [si , ti+1 ],

Lw (kxkP C + MU kBi kC kxkP C + Mf (1 + kxkP C )(ti+1 − si )) .

we have

kusi ,x (t)k ≤ Lw (1 + MU

(7) max

i=1,2,··· ,N

kBi kC + Mf )

max

i=1,2,··· ,N

(ti+1 − si )(1 + kxkP C ).

So we can known that For i = 1, 2, · · · , N , by H¨ older inequality, we have kx(t, z)k

≤

Z t



kU (t, si )zk + U (t, s)[f (s, x(s, z)) + B(s)u (s)]ds si ,x

si

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≤

MU kzk + MU

Z

t

si

√ Mf (1 + kx(s, z)k)ds + MU kBkC T kukL2 .

By Gronwall inequality, kx(t, z)k

≤ MU



kzk + Mf

max

i=1,2,··· ,N

√ (ti+1 − si ) + kBkC T kukL2



eMU Mf t .

kPi (z)k

= kx(ti+1 , z)k  ≤ MU kzk + Mf

max

i=1,2,··· ,N

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Hence

= ≤ ≤

kBi (si )Pi−1 (z)k  kBi kC MU kzk + Mf

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As a sequence, k(Gi ◦ Pi−1 )(z)k

max

i=1,2,··· ,N

a(i) + b(i)kzk.



eMU Mf ti+1 .



eMU Mf ti+1

√ (ti+1 − si ) + kBkC T kukL2

√ (ti+1 − si ) + kBkC T kukL2

M

Repeating the similar procedure, one can get to the formula (6).

Remark 2.6. Define P = GN ◦ PN −1 ◦ GN −1 ◦ PN −2 ◦ · · · ◦ G1 ◦ P0 . In Lemma 2.5, set j = N , we

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have

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kP (z)k ≤ a(1 + b + · · · + bN −1 ) + bN kzk,

CE

where a

:=

AC

b :=

max

i=1,2,··· ,N

max

i=1,2,··· ,N

kBi kC MU eMU Mf T ,  kBi kC MU Mf max

i=1,2,··· ,N

√

(ti+1 − si ) + kBkC T kuk

Remark 2.7. Note a(j) ≤ a, b(j) ≤ b, j = 1, 2, · · · , N , we have kGj ◦ Pj−1 ◦ Gj−1 ◦ Pj−2 ◦ · · · ◦ G1 ◦ P0 (z)k ≤ a(1 + b + · · · + bj−1 ) + bj kzk.

10

L2



eMU Mf T .

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Remark 2.8. By Remark 2.7, we have kPj−1 ◦ Gj−1 ◦ Pj−2 ◦ · · · ◦ G1 ◦ P0 (z)k  ≤ MU kGj−1 ◦ Pj−2 ◦ · · · ◦ G1 ◦ P0 (z)k +Mf max (ti+1 − si ) + kBkC T kuk i=1,2,··· ,N  ≤ MU a(1 + b + · · · + bj−2 ) + bj−1 kzk +Mf

max

i=1,2,··· ,N

L2



eMU Mf ti+1

CR IP T

√

 √ (ti+1 − si ) + kBkC T kukL2 eMU Mf T .

Now using the control function u(t) =

PN

i=0

usi ,x (t)χ[si ,ti+1 ] (t), t ∈ [0, T ] (usi ,x is defined in (2)),

we show that a nonlinear operator F : P C([0, T ], Rn1 ) → P C([0, T ], Rn1 ) given by

has a fixed point.

ED

M

AN US

 Rt   U (t, 0)x(0) + 0 U (t, s)[f (s, x(s)) + B(s)u(s)]ds, t ∈ [0, t1 ]      B (t)x(t− ), t ∈ (t , s ), i = 1, 2, · · · , N, i i i i (F x)(t) = Rt +   U (t, si )x(si ) + si U (t, s)[f (s, x(s)) + B(s)u(s)]ds,      t ∈ [s , t ], i = 1, 2, · · · , N, i i+1  Rt   U (t, 0)x(0) + 0 U (t, s)[f (s, x(s)) + B(s)u(s)]ds, t ∈ [0, t1 ]       B (t)(Pi−1 ◦ Gi−1 ◦ Pi−2 ◦ · · · ◦ G1 ◦ P0 )(x(0)), t ∈ (ti , si ),   i = i = 1, 2, · · · , N,      U (t, si )(Gi ◦ Pi−1 ◦ Gi−1 ◦ Pi−2 ◦ · · · ◦ G1 ◦ P0 )(x(0))      + R t U (t, s)[f (s, x(s)) + B(s)u(s)]ds, t ∈ [s , t ], i = 1, 2, · · · , N, i i+1 si

PT

Next, we rewrite F as follows:

AC

CE

 Rt   U (t, 0)x(0) + 0 U (t, s)(f (s, x(s)) + B(s)W −1 [x(t1 )    Rt    −U (t1 , 0)x(0) − 0 1 U (t1 , se)f (e s, x(e s))de s](s))ds, t ∈ [0, t1 ]       B (t)(Pi−1 ◦ Gi−1 ◦ Pi−2 ◦ · · · ◦ G1 ◦ P0 )(x(0)), t ∈ (ti , si ),   i (F x)(t) = i = 1, 2, · · · , N,     U (t, s )(G ◦ P  i i i−1 ◦ Gi−1 ◦ Pi−2 ◦ · · · ◦ G1 ◦ P0 )(x(0))    R  t   + si U (t, s)(f (s, x(s)) + B(s)W −1 [x(ti+1 ) − U (t, si )Bi (si )x(t−  i )     − R ti+1 U (t, se)f (e s, x(e s))de s](s))ds, t ∈ [si , ti+1 ], i = 1, 2, · · · , N. si

(8)

Clearly, (F x)(T ) = x1 , which means that the control u steers the system (1) from the initial state x0 to x1 in time T provided we obtain a fixed point of the nonlinear operator F . For each number k > 0, define Bk := {x ∈ P C([0, T ], Rn1 ) : kx(t)k ≤ k, t ∈ [0, T ]}. Then, there

is a bounded, closed, and convex subset of Bk in P C([0, T ], Rn1 ). 11

ACCEPTED MANUSCRIPT

Lemma 2.9. Assume [U1], [B1], [C1]-[C4] are satisfied. There exists a K > 0 such that F BK ⊆ BK . Proof. Let x ∈ BK , if t ∈ [0, t1 ], we derive kF x(t)k

Z

t

kU (t, s)[f (s, x(s)) + B(s)W −1 [x(t1 ) Z t1 −U (t1 , 0)x(0) − U (t1 , se)f (e s, x(e s))de s](s)]kds kU (t, 0)x0 k +

0

0

≤

CR IP T

≤

MU kx0 k + MU Mf t(1 + K) + MU kBkC Lw [K + MU kx0 k + MU Mf t1 (1 + K)]t

:= r1 (t) ≤ r1 (t1 ) := ρ1 K + M1∗ .

If t ∈ [si , ti+1 ], i = 1, 2, · · · , N , we have kF x(t)k

≤

AN US

If t ∈ (ti , si ), i = 1, 2, · · · , N , we have kF x(t)k ≤ kBi kC K.

MU kBi kC K + MU Mf (1 + K)(t − si ) + MU kBkC Lw [K + MU kBi kC K +MU Mf (1 + K)(ti+1 − si )](t − si )

M

∗ := ri+1 (t) ≤ ri+1 (ti+1 ) := ρi+1 K + Mi+1 ,

By [C4], we have F BK ⊆ BK . The proof is completed.

ED

Lemma 2.10. Assume [U1], [B1] and [C1]-[C4] hold. The operator F is continuous. Proof. Let {xn } be a sequence in BK such that kxn − xkP C([0,T ],Rn1 ) → 0 as n → ∞. By the

PT

conditions [C2] and [H1], for each t ∈ [0, T ], we have f (t, xn (t)) → f (t, x(t)), as n → ∞, and kf (t, xn (t)) − f (t, x(t))k ≤ Lf kxn (t) − x(t)k ≤ Lf kxn − xkP C([0,T ],Rn1 ) ≤ 2Lf K.

CE

By (7) and kukL2 ≤

√

T maxt∈[0,T ]

P

N i=0

 usi ,x (t)χ[si ,ti+1 ] (t) , we have kun −ukL2 → 0, as xn →

x, where un , u are the input of xn , x respectively.

AC

Case 1. If t ∈ [0, t1 ], we have kF xn (t) − F x(t)k

≤

Z t MU kxn − xkP C + MU kf (s, xn (s)) − f (s, x(s))kds 0 √ +MU kBkC T kun − ukL2 → 0, as n → ∞.

Case 2. If t ∈ (ti , si ), i = 1, 2, · · · , N , using Remark 2.7, we have kF xn (t) − F x(t)k ≤

kBi kC [(Pi−1 ◦ Gi−1 ◦ Pi−2 ◦ · · · ◦ G1 ◦ P0 )(xn (0)) 12

ACCEPTED MANUSCRIPT

−(Pi−1 ◦ Gi−1 ◦ Pi−2 ◦ · · · ◦ G1 ◦ P0 )(x(0))] i i−1 ≤ a(1 + bn + · · · + bi−1 ) − bi kx(0)k n ) + bn kxn (0)k − a(1 + b + · · · + b

≤ (bn − b) + · · · + (bi−1 − bi−1 ) + (bin − bi )kxn kP C + bi kxn − xkP C , n where max

i=1,2,··· ,N

max

√

i=1,2,··· ,N

(ti+1 − si ) + kBkC T kun kL2



eMU Mf T .

CR IP T

bn =

 kBi kC MU Mf

For kun kL2 − kukL2 ≤ kun − ukL2 , we have bn → b, as n → ∞. Thus, kF xn (t) − F x(t)k → 0 as n → ∞, in t ∈ (si , ti ).

Case 3. If t ∈ [si , ti+1 ], i = 1, 2, · · · , N , then one can repeat the similar computation in the above two cases to derive that F is continuous.

AN US

Lemma 2.11. Assume [U1], [B1] and [C1]-[C4] are satisfied. Then the set {F x : x ∈ BK } is compact.

Proof. We first show that F maps bounded set BK into equicontinuous family. Let x ∈ BK and 0 < τ1 < τ2 < t1 , kF x(τ2 ) − F x(τ1 )k

ED

0

kU (τ2 , 0) − U (τ1 , 0)kkx(0)k + (τ2 − τ1 )MU Mf (1 + K)

PT

≤

M

Z τ2 kU (τ2 , s)kMf (1 + kx(s)k)ds ≤ kU (τ2 , 0) − U (τ1 , 0)kkx(0)k + τ1 Z τ1 Z τ2 + kU (τ2 , s) − U (τ1 , s)kMf (1 + kx(s)k)ds + kU (τ2 , s)B(s)kku0,x (s)kds 0 τ1 Z τ1 + kU (τ2 , s) − U (τ1 , s)kkB(s)kku0,x (s)kds √ + sup kU (τ2 , s) − U (τ1 , s)kMf (1 + K)τ1 + (τ2 − τ1 )MU kBkC T kukL2 s∈[0,τ1 ]

CE

√ + sup kU (τ2 , s) − U (τ1 , s)kkBkC T kukL2 .

(9)

s∈[0,τ1 ]

AC

If ti < τ1 < τ2 < si , i = 1, 2, · · · , N , using Remark 2.8, we have kF x(τ2 ) − F x(τ1 )k

≤

kBi (τ2 ) − Bi (τ1 )kk(Pi−1 ◦ Gi−1 ◦ Pi−2 ◦ · · · ◦ G1 ◦ P0 )(x(0))k

≤

kBi (τ2 ) − Bi (τ1 )ka(1 + b + · · · + bj−1 ) + bj kx0 k

≤ kBi (τ2 ) − Bi (τ1 )kMU a(1 + b + · · · + bj−2 ) + bj−1 kx0 k √
+Mf max (ti+1 − si ) + kBkC T kukL2 eMU Mf T . i=1,2,··· ,N

13

(10)

ACCEPTED MANUSCRIPT

If si < τ1 < τ2 < ti+1 , i = 1, 2, · · · , N , using Remark 2.7, we have kF x(τ2 ) − F x(τ1 )k kU (τ2 , si ) − U (τ1 , si )kk(Gi ◦ Pi−1 ◦ Gi−1 ◦ Pi−2 ◦ · · · ◦ G1 ◦ P0 )(x(0))k Z τ2 + kU (τ2 , s)kMf (1 + kx(s)k)ds τ Z 1τ1 kU (τ2 , s) − U (τ1 , s)kMf (1 + kx(s)k)ds + Z0 τ2 + kU (τ2 , s)B(s)kkusi ,x (s)kds τ1 Z τ1 kU (τ2 , s) − U (τ1 , s)kkB(s)kkusi ,x (s)kds + 0

≤

CR IP T

≤

kU (τ2 , si ) − U (τ1 , si )kk(a(1 + b + · · · + bi−1 ) + bi kx0 k)

+(τ2 − τ1 )MU Mf (1 + K) + sup kU (τ2 , s) − U (τ1 , s)kMf (1 + K)τ1 s∈[0,τ1 ]

√ +(τ2 − τ1 )MU kBkC T kukL2 + sup kU (τ2 , s) − U (τ1 , s)kkBkC T kukL2 .

AN US

√

(11)

s∈[0,τ1 ]

Note that kBi (τ2 ) − Bi (τ1 )k tends to zero, and kU (τ2 , si ) − U (τ1 , si )k, and sups∈[0,τ1 ] kU (τ2 , s) − U (τ1 , s)k tend to zero as τ2 → τ1 in the uniformly operator topology. Thus, the right-hand sides of (9), (10) and (11) tend to zero as τ2 → τ1 . Then F maps BK into an equicontinuous family

M

of functions. Moreover, the family F BK is uniformly bounded due to Lemma 2.9. It follows the Arzela-Ascoli theorem that F BK is sequentially compact in P C([0, T ], Rn1 ).

ED

Theorem 2.12. Assume [H1], [B1], [U1] and [C1]-[C4] hold. System (1) is controllable on [0, T ]. Proof. It follows Lemmas 2.9, 2.10 and 2.11 that F BK is compact in P C([0, T ], Rn1 ) and F : BK →

PT

F BK ⊆ BK is a continuous operator. By Schauder’s fixed point theorem, F has a fixed point in F BK . Thus the system (1) is controllable on [0, T ].

CE

3. Existence of optimal controls

AC

In this section, we study optimal control problems for the current impulsive systems. Consider  SN   x(t) ˙ = A(t)x(t) + f (t, x(t)) + B(t)u(t), t ∈ j=0 [si , ti+1 ],      x(t) = B (t)x(t− ), t ∈ (ti , si ), i = 1, 2, · · · , N, i i + −   x(si ) = x(si ), i = 1, 2, · · · , N,      y(t) = C(t)x(t) + D(t)u(t), t ∈ [0, T ],

(12)

where C(·), D(·) ∈ C([0, T ], Rn1 ×n1 ), y(t) ∈ Rn1 .

Let x be the solution of system (12) corresponding to the control u ∈ Uad , where Uad denotes an

admissible control set belong to L2 ([0, T ], Rn1 ). We assume that Uad 6= Ø. 14

ACCEPTED MANUSCRIPT

Consider the following optimal control problem. OPT Problem: Find a u ∈ Uad , such that J(x, u) ≤ J(x, u) for all u ∈ Uad , x ∈ P C([0, T ], Rn1 ), where

0

T 2

ky(t) − yd (t)k dt =

Z

T

kC(t)x(t) + D(t)u(t) − yd (t)k2 dt,

0

and yd ∈ P C([0, T ], Rn1 ) is a given piecewise function.

CR IP T

J(x, u) =

Z

Denote F (t, x(t), u(t)) = A(t)x(t)+f (t, x(t))+B(t)u(t), F0 (t, x(t), u(t)) = kC(t)x(t)+D(t)u(t)−

yd (t)k2 , F1 (t, x(t), u(t)) = f (t, x(t)) + B(t)u(t).

AN US

We impose the following assumptions:

[O1] Suppose Uad ⊂ L2 ([0, T ], Rn1 ) is nonempty, bounded, closed and convex. [O2] F1 : [0, T ] × Rn1 × Rn1 → Rn1 is measurable in t on [0, T ] and there exist a constant MF1 > 0, such that

kF1 (t, x, u)k ≤ MF1 (1 + kxk).

M

(For the conditions of [B1], [C3] and [O1], we known that the constant MF1 is existence.)

ED

Theorem 3.1. Assume [H1], [U1], [B1], [O1] and [O2] hold. Then OPT Problem has a solution. Proof. If inf J(x, u) : u ∈ Uad = +∞, then the proof is trial. Without loss of generality, we assume that inf{J(x, u) : u ∈ Uad } < +∞. By the definition of

PT

J(·, ·), we have inf{J(x, u) : u ∈ Uad } ≥ 0. By definition of infimum, there exists a minimizing sequence {uk } ⊆ Uad , suppose xk is the solution of corresponding to uk , such that J(xk , uk ) → inf{J(x, u) : u ∈ Uad }, as k → ∞. Uad is

CE

bounded, it is clear that the sequence {uk } is bounded in L2 ([0, T ], Rn1 ). L2 ([0, T ], Rn1 ) is reflexive

Banach space, then there exist a subsequence, relabeled as {uk }, and u ∈ L2 ([0, T ], Rn1 ) such that

AC

uk * u in L2 ([0, T ], Rn1 ). Since Uad is closed and convex, by Mazur’s lemma, u ∈ Uad .

By Remark 2.6, we find that x is uniformly bounded. Next, recall the proof in Lemma 2.11, we

know that x is equicontinuity. Since x is uniformly bounded and equicontinuity, we have xk → x in P C([0, T ]). Then we have Rt − xk (s− i ) → x(si ) and si U (t, s)[f (s, xk (s)) − f (s, x(s))]ds → 0, i = 0, 1, 2, · · · , N .

Next, we check that the x is the solution of corresponding to u. Since U (t, ·) is compact, we have Z

t

si

U (t, s)B(s)(uk (s) − u(s))ds → 0, in P C([0, T ]), as uk * u.

15

ACCEPTED MANUSCRIPT

Note that xk (t)

=

U (t, si )xk (s− i )

Z

+

t

U (t, si )[f (s, xk (s)) + B(s)uk (s))]ds, t ∈ [si , ti+1 ],

si

= Bi (t)xk (t− i ), t ∈ [ti , si ].

xk (t) Thus, we obtain

x(t)

= =

U (t, si )x(s− i )+

Z

t

si

CR IP T

x(t)

U (t, si )[f (s, x(s)) + B(s)u(s))]ds, t ∈ [si , ti+1 ]

Bi (t)x(t− i ), t ∈ [ti , si ].

Finally, we show that J(·, ·) gets the minimum.

It is clearly that F0 (·, x(·), u(·)) is lower semi-continuous function about x and weakly semi-

by using Fatou’s Lemma, we have J(x(·), u(·))

≤

Z

AN US

continuous function about u. Then F0 (t, x(t), u(t)) ≤ lim inf k→∞ F0 (t, xk (t), uk (t)). For t ∈ [0, T ],

0

T

lim inf F0 (t, xk (t), uk (t))dt k→∞

≤ lim inf k→∞

T

F0 (t, xk (t), uk (t))dt

0

inf J(xk (·), uk (·)).

M

=

Z

ED

So J attains its minimum at u ∈ Uad . The proof is finished. 4. ILC problems for solving optimal control

PT

In this section, we give another numerical approach to solving optimal control problems with the help of ILC updating laws, which is easily to use computer to realize it.

AC

CE

Consider the following iterative impulsive systems:  SN   x˙ k (t) = A(t)xk (t) + f (t, xk (t)) + B(t)uk (t), t ∈ j=0 [si , ti+1 ],      x (t) = B (t)x (t− ) + w (t), t ∈ (t , s ), i = 1, 2, · · · , N, k i k i k i i + −   xk (si ) = xk (si ), i = 1, 2, · · · , N,      y (t) = C(t)x (t) + D(t)u (t), t ∈ [0, T ], k k k

(13)

where k denotes the iterative times, w ∈ C([0, T ], Rn1 ), xk (t), yk (t), uk (t) ∈ Rn1 . In general, xk (·) denotes the state, and uk (·) and yk (·) denote the control input and output, respectively. Denote the tracking error by ek (t) = yd (t) − yk (t). 16

(14)

ACCEPTED MANUSCRIPT

Denote ∆xk := xk+1 − xk , ∆uk := uk+1 − uk and ∆wk := wk+1 − wk . We impose the following assumptions. [H2] Let wk ∈ C([0, T ], Rn1 ) and there exists a w > 0 such that kwk (si )k ≤ w, i = 1, 2, · · · , N, ∀k ∈ N+ .

CR IP T

Linking system (13), we consider the following P -type updating law with initial state learning:   u k+1 (t) = uk (t) + Kp (t)ek (t), t ∈ [0, T ],  x (0) = x (0) + Le (0), k+1

k

k

(15)

where L, Kp (t), t ∈ [0, T ] are unknown parameters (matrix) to be determined and Kp (·) ∈

AN US

C([0, T ], Rn1 ×n1 ).

Theorem 4.1. For the system (13) and the reference trajectories yd . Assumptions [H1], [H2], [B1] and [U1] are satisfied. If we applying (15) to (13), then lim sup kek kλ k→∞

max

M

i=1,2,··· ,N

 kBi (·)kC }MU 2w 1 +

max

i=1,2,··· ,N

1 − kI − D(·)Kp (·)kC

kI − C(0)L − D(0)Kp (0)k < 1,

(17)

kI − D(·)Kp (·)kC < 1,

(18)

PT

holds provided that

 MU kBi (si )k eMU Lf T

ED

≤

kC(·)kC max{1,

CE

where I denotes identity matrix.

AC

Proof. Linking (14), (15) and (13), we have ek+1 (0)

(16)

= yd (0) − yk (0) − (yk+1 (0) − yk (0)) = ek (0) − (C(0)∆xk (0) + D(0)∆uk (0)) = ek (0) − (C(0)∆xk (0) + D(0)Kp (0)ek (0)) = (I − C(0)L − D(0)Kp (0))ek (0),

which implies that kek (0)k ≤ kI − C(0)L − D(0)Kp (0)kkek (0)k. 17

ACCEPTED MANUSCRIPT

It follows (17) and the contraction mapping principle that lim kek (0)k = 0.

(19)

k→∞

= yd (t) − yk+1 (t)

ek+1 (t)

CR IP T

For any t ∈ [0, T ], note (15) and (13), we have

= yd (t) − yk (t) + yk (t) − yk+1 (t)

= ek (t) − (C(t)∆xk (t) + D(t)∆uk (t))

= ek (t) − (C(t)∆xk (t) + D(t)Kp (t)ek (t)) (I − D(t)Kp (t))ek (t) − C(t)∆xk (t).

=

kek+1 (t)k

AN US

Taking the standard norm for both sides of (20), we have

(20)

≤

kI − D(t)Kp (t)kkek (t)k + kC(t)kk∆xk (t)k.

k∆xk (t)k = kU (t, si )∆xk (s+ i )+





U (t, s)(f (s, xk+1 (s)) − f (s, xk (s)) + B(s)∆uk (s))dsk

si

t

si

kU (t, s)k (Lf k∆xk (s)k + kB(s)Kp (s)ek (s)k) ds

k∆xk (s+ i )k +

max

t∈[si ,ti+1 ]

kB(t)Kp (t)k

PT

≤ MU

Z

t

ED

≤ MU k∆xk (s+ i )k +

Z

M

For any t ∈ [si , ti+1 ], using the solution of (13), we have

≤ MU

k∆xk (s+ i )k

Z

t

si

kek (s)kds + Lf

1 + max kB(t)Kp (t)k kek kλ eλt + Lf λ t∈[si ,ti+1 ]

Z

0

Z

t

si

t

 k∆xk (s)kds

 k∆xk (s)kds .

(21)

AC

CE

Case 1: if t ∈ [0, t1 ], then

k∆xk (0)k = kxk+1 (0) − xk (0)k = kLek (0)k.

(22)

By Gronwall inequality, we have k∆xk (t)k ≤ a0 (t) + b0

Z

0

t

k∆xk (s)kds ≤ a0 (t)eb0 t ≤ a0 (t)eMU Lf t1 ,

where a0 (t)

1 := MU k∆xk (0)k + MU max kB(t)Kp (t)k kek kλ eλt . λ t∈[0,t1 ]

18

(23)

ACCEPTED MANUSCRIPT

Combining (20), (21), (22) and (23), we have kek+1 (t)k

kI − D(t)Kp (t)kkek (t)k + kC(t)ka0 (t)eMU Lf t1 .

≤

(24)

Multiplying the factor e−λt on both sides of (24) and taking λ-norm, we obtain ≤

max kI − D(t)Kp (t)kkek kλ   1 + max kC(t)k MU kLkkek (0)k + O( )kek kλ eMU Lf t1 , λ t∈[0,t1 ] t∈[0,t1 ]

CR IP T

kek+1 kλ

where

(25)

1 1 O( ) := MU max kB(t)Kp (t)k . λ λ t∈[0,t1 ]

AN US

Linking (18) and (19) and taking sufficiently large λ for formula (25), we have lim kek kλ = 0.

k→∞

(26)

Case 2: if t ∈ [si , ti+1 ], i = 1, 2, · · · , N , applying [64, Lemma 4.2] to (21), we have k∆xk (t)k

≤ ai (t) + MU Lf

Z

0

t

k∆xk (s)kds + ζi k∆xk (t− i )k

M

≤ ai (t)(1 + ζi )eMU Lf t

(27)

ED

≤ ai (t)(1 + ζi )eMU Lf T , where

1 := MU k∆wk (si )k + O( )kek kλ eλt , λ ζi := MU kBi (si )k, 1 1 O( ) := MU max kB(t)Kp (t)k . λ λ t∈[si ,ti+1 ]

CE

PT

ai (t)

AC

Keeping in mind of the expression (20), (21) and (27), we have kek+1 (t)k

≤

kI − D(t)Kp (t)kkek (t)k + kC(t)kai (t)(1 + ζi )eMU Lf T .

(28)

Multiplying e−λt on both sides of (28) and taking λ-norm, we obtain kek+1 kλ

≤

kI − D(t)Kp (t)kkek kλ   1 + max kC(t)k MU k∆wk (si )k + O( )kek kλ (1 + ζi )eMU Lf T . λ t∈[si ,ti+1 ] max

t∈[si ,ti+1 ]

19

(29)

ACCEPTED MANUSCRIPT

Note (18) and (19), taking sufficiently large λ for (29) and using [71, Lemma 3], we have lim sup kek+1 kλ ≤ k→∞

di , 1 − ρi

(30)

where :=

ρi

:=

max

kC(t)kMU 2w(1 + ζi )eMU Lf T ,

max

kI − D(t)Kp (t)k.

t∈[si ,ti+1 ] t∈[si ,ti+1 ]

CR IP T

di

Case 3: if t ∈ [ti , si ], i = 1, 2, · · · , N , using the solution of (13), we have =

kek (t) − (yk+1 (t) − yk (t))k

=

k(I − D(t)Kp (t))ek (t) − C(t)Bi (t)(∆xk (t− i ))k

≤

kI − D(t)Kp (t)kkek (t)k + kC(t)Bi (t)kk∆xk (t− i )k.

AN US

kek+1 (t)k

For t− i ∈ [si−1 , ti ], i = 1, 2, · · · , N and using (23) and (27), we have ≤

kI − D(t)Kp (t)kkek (t)k + kC(t)Bi (t)kai−1 (t)(1 + ζi−1 )eMU Lf T .

(31)

M

kek+1 (t)k where ζ0 = 0.

≤

kI − D(·)Kp (·)kC kek kλ   1 +kC(·)Bi (·)kC MU k∆wk (si−1 )k + O( )kek kλ (1 + ζi−1 )eMU Lf T . λ

(32)

PT

kek+1 kλ

ED

Multiplying e−λt on both sides of (31) and taking λ-norm, we obtain

Linking (18), taking sufficiently large λ for (32), we have

CE


kC(·)Bi (·)kC MU k∆wk (si−1 )k + O( λ1 )kek kλ (1 + ζi−1 )eMU Lf T lim sup kek kλ ≤ . 1 − kI − D(·)Kp (·)kC k→∞

(33)

AC

By (26), (30) and (33), we derive (16). The proof is completed. To end this section, we put an additional condition on wk to derive another interesting result.

[H20 ] Suppose wk ∈ C([0, T ], Rn1 ) and lim

N P

k→∞ i=1

kwk (si )k = 0.

Corollary 4.2. For the system (13) and the reference trajectories yd . Assumptions [U1], [B1], [H1] and [H2’] are satisfied. If we apply (15) to (13), then limk→∞ kek kλ = 0 holds provided that the conditions (17) and (18) are satisfied.

20

ACCEPTED MANUSCRIPT

5. Random case In this section, we design the random ILC updating laws. Inspired by [63, Definition 1], throughout of this paper, we denote E{X} by the expectation of the stochastic variable X and p[g] by the occurrence probability of the event g. By Jensen’s inequality [65, Theorem 1.6.2], one has |E{X}| ≤ E{|X|}.

CR IP T

Define θ(i) by a stochastic variable in the ith phases. Let θ(i), i ∈ {0, 1, 2, 3, · · · , N } be a stochastic variable satisfying Bernoulli distribution and taking values 0 or 1.

Motivated by S. Liu et al. [62, Section 2], we define the sets γD (i) and γA (i) as follows:

  t , s
⊂ 0, T , i i γA (i) =  ∅,

Qi

Qi

j=0

θ(j) = 1, i ∈ {0, 1, 2, 3, · · · , N },

j=0

θ(j) = 0, i ∈ {0, 1, 2, 3, · · · , N },

Qi

j=0

θ(j) = 1, i ∈ {1, 2, 3, · · · , N },

j=0

θ(j) = 0, i ∈ {1, 2, 3, · · · , N },

AN US

and

   s , t  ⊂ 0, T , i i+1 γD (i) =  ∅,

Qi

where ti and si are satisfying 0 = s0 < t1 < s1 < · · · < sN −1 < tN < sN < tN +1 = T . Consider a class of random noninstantaneous impulsive differential equations described as:

i

(34)

ED

i

M

 SN   x(t) ˙ = A(t)x(t) + f (t, x(t)) + B(t)u(t), t ∈ j=0 γD (j),   T SN x(t) = Bi (t)x(t− ) + w(t), t ∈ (ti , si ) ( i=1 γA (i)), i     x(s+ ) = x(s− ),

e, N e := min{i, γA (i) = ∅} − 1, A, B, Bi ∈ C([0, T ], Rn1 ×n1 ), f ∈ C([0, T ] × where i = 1, 2, · · · , N

PT

Rn1 , Rn1 ), w ∈ C([0, T ], Rn1 ) and x(t) ∈ Rn1 , t ∈ [0, T ].

One can use standard methods to derive that the problem (34) has a unique solution x ∈

CE

P C([0, T ], Rn1 ) given by x(t)

AC

x(t)

x(s+ i )

= U (t, si )x(si ) +

Z

t

si

U (t − s)[f (s, x(s)) + B(s)u(s)]ds, t ∈ γD (i) 6= ∅,

=

Bi (t)x(t− i ) + w(t), t ∈ γA (i) 6= ∅,

=

x(s− i ), i = 1, · · · , N.

In order to make the reader to understand the solutions to the problem (34), we list the following

figures. In the illustration, the differential equation is described by a solid line (the graph area is yellow) and the algebraic equation is described by a dashed line (the graph area is grey). We divide the interval [0, T ] into N + 1 smaller intervals according to impulsive time (ti , si ) in the problem (34). Start from second time periods ((ti , ti+1 ], i = 1, 2, · · · , N ), every smaller interval has two parts. 21

CR IP T

ACCEPTED MANUSCRIPT

(b) Running at [0, t3 ] and stop at t3 .

AN US

(a) Running at all subintervals.

(c) Running at [0, t2 ], stop at (t2 , t3 ], happening at (t3 , T ].

(d) Running at [0, t2 ], stop at (t2 , t3 ], happening at (t3 , t4 ], stop at (t4 , T ].

Part one: the noninstantaneous impulsive characterization by the algebraic equation in interval

M

(ti , si ).

Part two: In times [si , ti+1 ], the trajectory of the system is depicted by the differential equation.

ED

We hope that the system will be able to choose to stop or continue to move to the time T before entering the ”noninstantaneous impulsive” phase. For every iteration, if the m-th (0 ≤ m < N ) segment fails to run, then we restart the process.

PT

That is to say, the m-th, m + 1-th, · · · , N -th subintervals will not happen. The system will start the next iteration process directly. Thus, the probability of the occurrence of each segment not only depends on own decision, but also on the previous step. If the first segment fails to run, the system

CE

will start the next iteration process directly and not record the number of this iteration. Thus, we set p[θ(0) = 1] = 1 without loss of the general.

AC

Definition a step function p(·) ∈ P C([0, T ], [0, 1]) by   1, t ∈ [0, t ], 1 p(t) =  Qi p[θ(j) = 1], t ∈ (t , t ], i = 1, · · · , N, i i+1 j=0

Obviously, 0 ≤ p(t) ≤ 1, t ∈ [0, T ]. Without loss of generality, we only consider 0 < p(t) ≤ 1, t ∈ [0, T ]. If p(t) = 0, t ∈ [0, T ] then the case is no longer meaningful. Before we design the new random learning law, we need to define the error in the sense of

22

ACCEPTED MANUSCRIPT

probability. In general, the tracking error is defined as the following formula in the traditional ILC: ek (t) = yd (t) − yk (t). Then one can apply control means (also known as learning law) of the error as ”experience” to update the controller so that the controller has the ability to learn. Since the absent signals are

CR IP T

unavailable, the error of traditional definition is not well defined in the interval [0, T ]. Therefore, we can not expect to be updated input as defined on [0, T ].

In order to update the learning law, we need to handle with the missing information in the error. Qi To achieve this aim, we make the discrete variable j=0 θ(j) to be a piecewise continuous variable by zero-order holder as follows:

Next, we define a correction tracking error:

AN US

  θ(0), t ∈ [0, t ], 1 ϑ(t) =  Qi θ(j), t ∈ (t , t ], i = 1, · · · , N, i i+1 j=0

M

  y (t) − y (t), t ∈ (SN γ (i)) S(SN γ (i)), d k i=0 D i=1 A ∗ ek (t) := ϑ(t)ek (t) =  0, others.

In addition, the correction tracking error can also be written:

ED

ek (t) = (yd (t) − yk (t))χ(SN

i=0

γD (i))

S SN ( i=1 γA (i)) (t).

where χ is characteristic function.

PT

Next we will using the correction tracking error to deign ILC scheme. Consider the following impulsive system with randomly varying trial lengths:

AC

CE

 SN   x˙ k (t) = A(t)xk (t) + f (t, xk (t)) + B(t)uk (t), t ∈ j=0 γD (j),      xk (t) = Bi (t)xk (t− ) + w(t), t ∈ γA (i), i = 1, 2, · · · , N, i + −   xk (si ) = xk (si ), i = 1, 2, · · · , min{i, γA (i) = ∅} − 1,          yk (t) = C(t)xk (t) + D(t)uk (t), t ∈ [0, t1 ] S SN γD (j) S SN γA (j) , j=1 j=1

(35)

where k denotes the iterative times, A, B, Bi ∈ C([0, T ], Rn1 ×n1 ), f ∈ C([0, T ] × Rn1 , Rn1 ), w ∈

C([0, T ], Rn1 ), xk (t), yk (t), uk (t) ∈ Rn1 . In general, xk (·) denotes the state, and uk (·) and yk (·) denote control input and output, respectively. e∗k (t) denotes the tracking error.

23

ACCEPTED MANUSCRIPT

We consider the P -type learning law:  ∗  u k+1 (t) = uk (t) + Kp (t)ek (t), t ∈ [0, T ],  x (0) = x (0) + Le (0). k+1

k

(36)

k

Now we are ready to state our main result.

and [H2] are satisfied. If we applying (36) to (35), then lim sup kE{e∗k }kλ

(37)

k→∞



kBi (·)kC }MU 2w 1 +

max

i=1,2,··· ,N

1 − kI − D(·)Kp (·)kC

holds provided that

max

i=1,2,··· ,N



MU kBi (si )k eMU Lf

AN US

≤

kC(·)kC max{1,

CR IP T

Theorem 5.1. For the system (35) and the reference trajectories yd . Assumptions [U1], [B1], [H1]

kI − C(0)L − D(0)Kp (0)k < 1,

(38)

kI − D(·)Kp (·)kC < 1.

(39)

M

Proof. Similar the formula (19), we have

lim kek (0)k = 0.

ED

k→∞

For any t ∈ [0, T ], note (36) and (35), we have =

ϑ(t)(yd (t) − yk+1 (t))

=

ϑ(t)(yd (t) − yk (t) + yk (t) − yk+1 (t))

=

ϑ(t)(ek (t) − (C(t)∆xk (t) + D(t)∆uk (t)))

=

ϑ(t)(ek (t) − (C(t)∆xk (t) + D(t)Kp (t)ek (t)))

=

ϑ(t)((I − D(t)Kp (t))ek (t) − C(t)∆xk (t)).

AC

CE

PT

e∗k+1 (t)

(40)

Applying the operator E{·} for both sides of (40), we obtain E{e∗k+1 (t)}

=

(I − D(t)Kp (t))E{e∗k (t)} − C(t)E{∆xk (t)}.

(41)

Taking the standard norm for both sides of (41), we have kE{e∗k+1 (t)}k

≤

kI − D(t)Kp (t)kkE{e∗k (t)}k + kC(t)p(t)kkE{∆xk (t)}k, 24

(42)

ACCEPTED MANUSCRIPT

where kp(t)k ≤ 1. Similar the formula (21), we obtain  ≤ MU k∆xk (s+ i )k +

1 kB(t)Kp (t)k ke∗k kλ eλt λ t∈[si ,ti+1 ]  Z t +Lf k∆xk (s)kds .

k∆xk (t)k

max

(43)

Case 1: if t ∈ [0, t1 ], by Gronwall inequality, we have a0 (t)eMU Lf t1 ,

k∆xk (t)k ≤ where

(44)

1 := MU k∆xk (0)k + MU max kB(t)Kp (t)k ke∗k kλ eλt . λ t∈[0,t1 ]

AN US

a0 (t)

CR IP T

0

Multiplying the factor e−λt and applying the operator E{·} on both side of (44), we have kE{∆xk }kλ

≤



MU k∆xk (0)k

(45)

ED

M

 1 +MU max kB(t)Kp (t)k kE{e∗k }kλ eMU Lf t1 λ t∈[0,t1 ]   1 ≤ MU k∆xk (0)k + O( )kE{e∗k }kλ eMU Lf t1 , λ

where

PT

1 O( ) λ

1 := MU max kB(t)Kp (t)k . λ t∈[0,t1 ]

CE

Taking the λ-norm for both sides of (42) and substituting (45) to it, we have

AC

kE{e∗k+1 }kλ

max kI − D(t)Kp (t)kkE{e∗k }kλ   1 ∗ + max kC(t)k MU k∆xk (0)k + O( )kE{ek }kλ eMU Lf t1 λ t∈[0,t1 ] ∗ = max kI − D(t)Kp (t)kkE{ek }kλ t∈[0,t1 ]   1 ∗ + max kC(t)k MU kLek (0)k + O( )kE{ek }kλ eMU Lf t1 . λ t∈[0,t1 ]

≤

t∈[0,t1 ]

e , by applying [64, Lemma 4.2] to (43), then Case 2: for t ∈ [si , ti+1 ], i = 1, · · · , N k∆xk (t)k

Z

t

k∆xk (s)kds + ζi k∆xk (t− i )k

≤

ai (t) + bi

≤

ai (t)(1 + ζi )eMU Lf t

0

25

(46)

ACCEPTED MANUSCRIPT

≤ ai (t)(1 + ζi )eMU Lf T ,

(47)

where 1 := MU k∆wk (si )k + O( )ke∗k kλ eλt , λ ζi := MU kBi (si )k, 1 1 O( ) := MU max kB(t)Kp (t)k . λ λ t∈[si ,ti+1 ]

CR IP T

ai (t)

Multiplying the factor e−λt and applying the operator E{·} on both side of (47), we have kE{∆xk }kλ

  1 MU k∆wk (si )k + O( )kE{e∗k }kλ (1 + ζi )eMU Lf T . λ

≤

(48)

kE{e∗k+1 }kλ

≤

AN US

Taking the λ-norm for both sides of (42), and substituting (48) to it, we have

kI − D(t)Kp (t)kkE{e∗k }kλ   1 + max kC(t)k MU k∆wk (si )k + O( )kE{e∗k }kλ (1 + ζi ) eMU Lf T . (49) λ t∈[si ,ti+1 ] max

t∈[si ,ti+1 ]

kE{e∗k+1 (t)}k =

M

e , using the solution of (35), we have Case 3: if t ∈ [ti , si ], i = 1, 2, · · · , N kE{e∗k (t)} − E{yk+1 (t) − yk (t)}k

k(I − D(t)Kp (t))E{ek (t)} − C(t)Bi (t)E{∆xk (t− i )}k

≤

kI − D(t)Kp (t)kkE{ek (t)}k + kC(t)Bi (t)kkE{∆xk (t− i )}k.

ED

=

PT

For t− i ∈ [si−1 , ti ], i = 1, 2, · · · , N and using (44) and (47), we have

CE

kE{e∗k+1 (t)}k

≤

kI − D(t)Kp (t)kkE{e∗k (t)}k + kC(t)Bi (t)kai−1 (t)(1 + ζi−1 )eMU Lf T .

(50)

AC

where ζ0 = 0.

Multiplying e−λt on both sides of (50) and applying λ-norm, we obtain kE{e∗k+1 }kλ

≤ kI − D(·)Kp (·)kC kE{e∗k }kλ   1 +kC(·)Bi (·)kC MU k∆wk (si−1 )k + O( )kE{e∗k }kλ (1 + ζi−1 )eMU Lf T . (51) λ

Linking (39), and taking sufficiently large λ for (46), (49) and (51), using [71, Lemma 3], we derive (37).

26

ACCEPTED MANUSCRIPT

Corollary 5.2. For the system (35) and the reference trajectories yd . Assumptions [U1], [B1] , [H1] and [H2’] are satisfied. If we apply (36) to (35), then limk→∞ kE{ek }kλ = 0 holds provided that the conditions (38) and (39) are satisfied.

CR IP T

6. Simulation examples In this section, we present some explicit examples to demonstrate the validity of theoretical results and designed algorithm methods. Example 6.1. Consider the following impulsive system:

AN US

     0.2 0 S    xk (t) + 0.1 sin(xk (t)) + 0.1uk (t), t ∈ 3 [si , ti+1 ], x˙ (t) =   i=0   k 0 0.1 S3   xk (t) = 0.1 cos(t − si )xk (t−  i ), t ∈ i=1 (ti , si ),     y (t) = 0.7x (t) + 0.5u (t), t ∈ [0, T ], k k k

(52)

where s0 = 0 < t1 = 1 < s1 = 2 < t2 = 3 < s2 = 3.5 < t3 = 4 < s3 = 4.5 < t4 = T = 5. Then Z

ti+1

si

eAs ds < u, en > en = A−1 (e(ti+1 −si )A ) < u, en > en .

M

Wi u =

ED

where {en } is the orthonormal set of L2 ([0, T ], R2 ). We have =

A−1 (e(ti+1 −si )A ),

Wi−1

=

(e(ti+1 −si )A )−1 A.

PT

Wi

For [U1], we have kU (t, s)k = keA(t−s) k ≤ keAT k = MU ≈ 2.7183. Then we have maxi=1,2,··· ,N +1 ρi ≈

CE

0.5699 < 1 and k sin(xk (t))k ≤ 1 + kxk (t)k. Thus, [H1], [U1], [B1], [C1]-[C4] are satisfied. Then the system (6.1) is controllable.

AC

We choose the learning law as follows:

uk+1 (t) xk+1 (0)



= uk (t) + 

0.5

0

0

0.5 sin( 5t + 0.1)

= xk (0) + 0.5ek (0). 

The initial state and the 1st control are proposed as x0 = 

27



 ek (t), t ∈ [0, 5],

−2 0





 and u1 (t) = 

(53) 0 0



 , t ∈ [0, 5],

ACCEPTED MANUSCRIPT

respectively. The reference trajectory is given as: 

yd (t) = 

yd1 (t) yd2 (t)



 , t ∈ [0, 5],

(54)

where yd1 (t) = sin(5t),

CR IP T

   0.48t2 (1 − 0.2t), t ∈ [0, 1],      0.48t2 (1 − 0.2t) + 1, t ∈ (1, 3], yd2 (t) =   0.48t2 (1 − 0.2t) − 1, t ∈ (3, 4],      0.48t2 (1 − 0.2t) + 2, t ∈ (4, 5].

AN US

We can see

kI − C(0)L − D(0)Kp (0)k ≈ 0.6250 < 1, kI − D(·)Kp (·)kC ≈ 0.9750 < 1. Then all the conditions of Corollary 4.2 are satisfied.

The upper and the middle figure of Figure 1 shows the system output yk of the 1, 5, 10, · · · , 100

M

iterations (blue lines) and the referenced trajectory yd (red stars). The grey area means the impulsive generator time. The lower figure shows the tracking error in each iteration. The tracking error at

AC

CE

PT

ED

the 100th iteration is 6.97 × 10−4 , which is very small.

Figure 1: The system output and the tracking error.

Example 6.2. We randomly generate a set of 0-1 sequences θk (i), i = 0, 1, 2, 3, k = 1, 2, · · · , 100. 28

ACCEPTED MANUSCRIPT

Consider the following system:      0.2 0 S    xk (t) + 0.1 sin(xk (t)) + 0.1uk (t), t ∈ 3 γD (i), x˙ k (t) =   i=0   0 0.1 S3   xk (t) = 0.1 cos(t − si )xk (t−  i ), t ∈ i=1 γA (i),   S S3 S3   yk (t) = 0.7xk (t) + 0.5uk (t), t ∈ ( i=0 γD (i)) ( i=1 γA (i)),

(55)

the learning law as follows:

uk+1 (t) xk+1 (0)



= uk (t) + 

0.5

0

0

0.5 sin( 5t + 0.1)

= xk (0) + 0.5ek (0).

 ek (t), t ∈ [0, 5], 

AN US





CR IP T

where s0 = 0 < t1 = 1 < s1 = 2 < t2 = 3 < s2 = 3.5 < t3 = 4 < s3 = 4.5 < t4 = T = 5. We choose



0



 and u1 (t) =   , t ∈ [0, 5], 0 0 respectively. The reference trajectory is given as (54). The upper and the middle figure of Figure 2

AC

CE

PT

ED

M

The initial state and the 1st control are proposed as x0 = 

−2

(56)

Figure 2: The system output and the tracking error.

shows the system output yk of the 1, 5, 10, · · · , 100 iterations (blue lines) and the referenced trajectory

yd (red stars). The grey area means the impulsive generator time. The lower figure shows the tracking error in each iteration. The tracking error of each iterative meet the Table 1.

29

ACCEPTED MANUSCRIPT

Table 1: Selected operation information

sita0

sita1

sita2

sita3

running subintervals

tracking error

1

1

1

1

0

3

3.8098

2

1

1

1

0

3

2.9802

3

1

0

1

1

1

1.3781

4

1

1

0

1

2

2.3488

5

1

1

0

1

2

1.8653

10

1

0

1

0

1

20

1

0

1

0

1

30

1

1

0

1

2

40

1

1

0

0

2

50

1

0

0

0

1

60

1

1

1

0

3

70

1

1

1

80

1

0

0

90

1

1

0

95

1

1

0

96

1

1

1

97

1

1

1

98

1

0

99

1

0

100

1

1

0.1865 0.0830 0.3465 0.2357 0.0090 0.1374

1

4

0.9707

1

1

0.0018

1

2

0.0742

1

2

0.0589

1

4

0.5012

0

3

0.0505

M

AN US

CR IP T

iterative time

0

1

0.0008

0

1

1

0.0007

1

0

3

0.0467

ED

1

Example 6.3. Consider the following system:

CE

PT

 S3   x˙ (t) = 0.5 cos(t)xk (t) + 0.1 sin(xk (t)) + cos(t)uk (t), t ∈ i=0 [si , ti+1 ],   k S3 xk (t) = 0.1 cos(t − si )xk (t− i ), t ∈ i=1 (ti , si ),     y (t) = 0.7x (t) + u (t), t ∈ [0, T ], k

k

(57)

k

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where s0 = 0 < t1 = 1 < s1 = 2 < t2 = 3 < s2 = 3.5 < t3 = 4 < s3 = 4.5 < t4 = T = 5. Then Wi u =

Z

ti+1

e

si

Rs

si

A(τ )dτ

ds < u, en > en

= 2(e0.5(sin ti+1 −sin si ) ) < u, en > en .

where {en } is the orthonormal set of L2 ([0, T ], R). We have Wi

=

Wi−1

=

2(e0.5(sin ti+1 −sin si ) ), 1 . 2(e0.5(sin ti+1 −sin si ) ) 30

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For [U1], we have kU (t, s)k = ke

Rs 0

A(τ )dτ

k ≤ ke0.5 sin(T ) k = MU ≈ 0.6191. Then maxi=1,2,··· ,N +1 ρi ≈

0.8393 < 1. Thus, [H1], [U1], [C1]-[C4] are satisfied. Then the system (57) is controllable. We choose the learning law as follows: =

xk+1 (0)

=

1 uk (t) + 0.5 sin( t + 0.1)ek (t), t ∈ [0, 5], 5 xk (0) + 0.5ek (0).

(58)

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uk+1 (t)

The initial state and the 1st control are proposed as x0 = −2 and u1 (t) = 0, respectively. The reference trajectory is given as:

   0.48t2 (1 − 0.2t), t ∈ [0, 1],      0.48t2 (1 − 0.2t) + 1, t ∈ (1, 3], yd (t) =   0.48t2 (1 − 0.2t) − 1, t ∈ (3, 4],      0.48t2 (1 − 0.2t) + 2, t ∈ (4, 5].

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(59)

We can see

kI − C(0)L − D(0)Kp (0)k ≈ 0.6001 < 1,

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kI − D(·)Kp (·)kC ≈ 0.9501 < 1. Then all the conditions of Corollary 4.2 are satisfied.

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The upper figure of Figure 3 shows the system output yk of the 1, 5, 10, · · · , 100 iterations (blue lines) and the referenced trajectory yd (red stars). The grey area means the impulsive generator time. The lower figure shows the tracking error in each iteration. The tracking error at the 100th

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iteration is 0.0084, which is very small.

Example 6.4. We randomly generate a set of 0-1 sequences θk (i), i = 0, 1, 2, 3, k = 1, 2, · · · , 100.

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Consider the following system:

 S3   x˙ (t) = 0.5 cos(t)xk (t) + 0.1 sin(x(t)) + cos(t)uk (t), t ∈ i=0 γD (i),   k S3 x(t) = 0.1 cos(t − si )x(t− i ), t ∈ i=1 γA (i),     y (t) = 0.7x (t) + u (t), t ∈ (S3 γ (i)) S(S3 γ (i)), k

k

k

i=0

D

i=1

(60)

A

where s0 = 0 < t1 = 1 < s1 = 2 < t2 = 3 < s2 = 3.5 < t3 = 4 < s3 = 4.5 < t4 = T = 5. We

choose the learning law as (58). The initial state and the 1st control are proposed as x0 = −2 and u1 (t) = 0, respectively. The reference trajectory is given as (59). The upper and the middle figure of Figure 4 shows the system output yk of the 1, 5, 10, · · · , 100 iterations (blue lines) and the referenced trajectory yd (red stars). The grey area means the impulsive generator time. The lower figure shows the tracking error in each iteration. The tracking error of 31

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Figure 3: The system output and the tracking error.

each iterative meet the Table 2. 7. Conclusions

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Controllability and optimal controls of system governed by a class of semilinear differential equations with noninstantaneous impulses via ILC problems for fixed and random batch length cases

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have been investigated. We turn controllability problem into a fixed point problem for a suitable operator, then contraction mapping principle and Schauder’s fixed point theorem are used to complete the proof. We present the existence of optimal controls problem for the above controlled systems.

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Further, we study the convergence results in determine case and extend it to randomly varying trial length case, which shows the real procedure in tacking problem. The theoretically results are illustrated by some numerical examples. ILC updating algorithm is successfully used to find the

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optimal control function to solve optimization problem via computer techniques. In the forthcoming papers, we will investigate optimal controls of time-fractional partial equations

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[66, 67, 68, 69] and fractional damped equations [70]. Acknowledgments The authors are grateful to the referees for their careful reading of the manuscript and valuable

comments. The authors thank the help from the editor too. References [1] D. D. Bainov, P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, CRC Press, 1993.

32

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Figure 4: The system output and the tracking error.

[2] D. D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, Series on Advances in Mathematics for Applied Sciences, vol. 28. Singapore: World Scientific, 1995.

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tions, 1998.

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[5] A. M. Samoilenko, N. A. Perestyuk, Y. Chapovsky, Impulsive Differential Equations, World Scientific, 1995. [6] M. Benchohra, J. Henderson, S. K. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi

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Publishing Corporation, 2006.

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Table 2: Selected operation information

θk (0)

θk (1)

θk (2)

θk (3)

running subintervals

tracking error

1

1

1

1

0

3

4.9359

2

1

1

1

0

3

4.1345

3

1

0

1

1

1

1.6417

4

1

1

0

1

2

3.4308

5

1

1

0

1

2

2.8265

10

1

0

1

0

1

0.8829

20

1

0

1

0

1

0.5263

30

1

1

0

1

2

0.3138

40

1

1

0

0

2

0.1870

50

1

0

0

0

1

0.1115

60

1

1

1

0

3

0.0665

70

1

1

1

80

1

0

0

90

1

1

0

95

1

1

0

96

1

1

1

97

1

1

1

98

1

0

99

1

0

100

1

1

4

0.1937

1

1

0.0236

1

2

0.0141

1

2

0.0109

1

4

0.0437

0

3

0.0098

1

0

1

0.0093

0

1

1

0.0088

1

0

3

0.0084

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1

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iterative time (k)

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