Approximate Controllability of semi-linear Heat equation with Non-instantaneous Impulses, Memory and Delay
AAPPROXIMATE CONTROLLABILITY OF SEMI-LINEAR HEATEQUATION WITH NON INSTANTANEOUS IMPULSES,MEMORY AND DELAY
H. LEIVA , W. ZOUHAIR AND M. N. ENTEKHABI Abstract.
The semilinear heat equation with non instantaneous impulses (NII) , memory and delay is considered and its approximate controllabilityis obtained. This is done by employing a technique that avoids fixed pointtheorems and pulls back the control solution to a fixed curve in a short timeinterval. We demonstrate, once again, that the controllability of a system isrobust under the influence of non instantaneous impulses, memory and delays.In support, a numerical example with simulation for a linear heat equation isgiven to validate the obtained controllability result for the linear part. Finallywe present some open problems and a possible general framework to study thecontrollability of non instantaneous impulsive (NII) semilinear equations.
Contents
1. Introduction 12. Abstract Formulation of the Problem 33. Approximate Controllability of the Linear Equation 54. Approximate Controllability of the Semilinear System (1.1) 75. Example 106. Final Remark 116.1. Open Problem 116.2. Open Problem 116.3. Open Problem 12References 121.
Introduction
The theory of impulsive differential equations was initiated by A. Mishkis andV. D. Mil’man in 1960, it has become an important area of investigation in recentyears. There are many practical examples of impulsive control systems, The growthof a population diffusing throughout its habitat, the amount of money in a stockmarket, the spread of an acoustic vibrations,we can find this notion in severalother fields like: neural networks, ecology, chemistery, biotechnology, radiophysics,theoretical physics, mathematical economy. we can easily seen in these examples
Mathematics Subject Classification.
Key words and phrases.
Semilinear Heat equation, Non instantaneous impulsive, ApproximateControllability, Evolution equation with memory and Delay . a r X i v : . [ m a t h . D S ] A ug H. LEIVA, W. ZOUHAIR AND M.N. ENTEKHABI where abrupt changes such as harvesting, disasters and instantaneous storage canoccur. These problems are modelled by impulsive differential equations.Modelling such real phenomena, we notice that the impulse started at a givenmoment and its action remains active for a finite time. Such an impulse is known asa non-instantaneous impulse (NII). This notion appears for the first time in 2012.After that, it has become an area of interest for many researchers. For more, werefer to our readers [7, 8, 9].The evolution of this theory was quite slow because of the complexity of ma-nipulating such equations. For example, if we consider systems with impulses (in-stantaneous or non-instantaneous). Since the impulses involve instantaneous anddiscontinuous changes at various time instants witch influencing the solutions, thiscan lead to instability (respectively uncontrollability) of the differential equationor inversely its stability (respectively controllability), (see [1], [2] ). Afterwards,many scientists contributed in the enrichment of this theory, they launched differ-ent studies on this subject and large number of results were established .Controllability is a mathematical problem, which consists of finding a controlssteering the system from an arbitrary initial state to a final state in a finite intervalof time, the controllability of the impulsive systems is studied by several authors(see [3, 4, 5, 6]). In this work we prove the interior approximate controllabilityfor the following non instantaneous impulsive (NII) semilinear heat equation withmemory and delay. ∂ω∂t + Aω = θ u ( t, x ) + (cid:90) t M ( t, s ) g ( ω ( s − r, x )) d s in ∪ Ni =0 ( s i , t i +1 ) × [0 , π ] , + f ( t, ω ( t − r, x ) , u ( t, x )) ,ω ( t,
0) = ω ( t, π ) = 0 , on (0 , T ) ,ω ( s, x ) = h ( s, x ) , in [ − r, × [0 , π ] ,ω ( t, x ) = G i ( t, ω ( t, x ) , u ( t, x )) , in ∪ Ni =0 ( t i , s i ] × [0 , π ] . (1.1)Where 0 = s = t < t ≤ s < ... < t N ≤ s N < t N +1 = T are fixed real numbers, h : [ − r, × [0 , π ] → R is a piecewise continuous function in s , f : [0 , T ] × R × R → R , X = L [0 , π ] and A : D ( A ) ⊂ X → X is the operator Aψ = − ψ xx with domain D ( A ) := { ψ ∈ X : ψ, ψ x absolutely continuous , ψ xx ∈ X , ψ (0) = ψ ( π ) = 0 } , and ( D ( A )) / = X / , θ is an open nonempty subset of [0 , π ] , θ denotes the char-acteristic function of the set θ. PPROXIMATE CONTROLLABILITY OF SEMI-LINEAR HEAT EQUATION 3
Example of the effect of non-instantaneous pulses (NII) . This paper is organized as follows. In section 2, we briefly present the problemformulation and related definition. In section 3 and 4, we discuss the approximationof controlability for the linear and and the semilinear system. The last section isdevoted to some related open problems and application.2.
Abstract Formulation of the Problem
We begin the section introducing notations and some hypothesis. Throughoutthis paper we use the following notations for (1.1):It is well known that − A : D ( A ) ⊂ X → X , is the generator of a Strongly continuousanalytic semigroup ( S ( t )) t ≥ on X . Moreover, the operator A and the semigroup( S ( t )) t ≥ can be represented as follows: Ax = ∞ (cid:88) n =1 λ n (cid:104) x, φ n (cid:105) φ n , x ∈ X , where λ n = n , φ n ( ξ ) = sin nξ and (cid:104)· , ·(cid:105) is the inner product in X . So, the stronglycontinuous semigroup ( S ( t )) t ≥ generated by A is also compact and given by S ( t ) x = ∞ (cid:88) n =1 e − n t (cid:104) x, φ n (cid:105) φ n , x ∈ X . As a consequence, we have the following estimate: (cid:107) S ( t ) (cid:107)≤ e − t , t ≥ . Consequently, systems (1.1) can be written as an abstract functional differentialequations with memory in X : ω (cid:48) = − Aω + B θ u + (cid:90) t M ( t, s ) g ( ω s ( − r ))d s + f ( t, ω t ( − r ) , u ( t )) , ∪ Ni =0 ( s i , t i +1 ) ,ω ( t ) = h ( t ) , [ − r, ,ω ( t ) = G i ( t, ω ( t ) , u ( t )) , ∪ Ni =0 ( t i , s i ] , (2.1) H. LEIVA, W. ZOUHAIR AND M.N. ENTEKHABI where U = X , u ∈ L ([0 , T ]; U ) , B θ : U −→ X is a bounded linear operator suchthat B θ u = θ u, ω t ∈ C ([ − r, X ) defined by ω t ( s ) = ω ( t + s ), and the functions g : L [0 , π ] −→ L [0 , π ] , G i : [ t i , s i ] × X × U −→ L [0 , π ] , f : [0 , T ] × PW × U −→ L [0 , π ] , are defined by g ( z )( x ) = g ( z ( x )) ,f ( t, ϕ, u )( x ) = f ( t, ϕ ( − r, x ) , u ( x )) ,G k ( t, z, u )( x ) = G k ( t, z ( x ) , u ( x )) for k = 1 , ..., N, where PW is the space of piecewise continuous functions: PW = (cid:110) h : [ − r, −→ X / : h is piecewise continuous (cid:111) , endowed with the norm (cid:107) h (cid:107) = max {(cid:107) h ( t ) (cid:107) X : − r ≤ t ≤ } . To defined the mild solution of the problem above, we introduce the followingfunction: f : [0 , T ] × PW × U → X , given by f ( t, X, U ) = (cid:90) t M ( t, s ) g ( X s ) d s + f ( t, X t , U ( t )) , then, from system (2.1), we obtain the following non-autonomous differential equa-tion with non-instantaneous impulses ω (cid:48) = − Aω + B θ u + f ( t, ω t , u ) , ∪ Ni =0 ( s i , t i +1 ] ,ω ( t ) = h ( t ) , [ − r, ,ω ( t ) = G i ( t, ω ( t ) , u ( t )) , ∪ Ni =0 ( t i , s i ] . (2.2)We consider the space PC ( X ) of all functions ϕ : [ − r, T ] −→ X such that ϕ ( · ) ispiecewise continuous on [ − r,
0] and continuous on [0 , T ] except at points t i wherethe side limits exist ϕ ( t − i ) = ϕ ( t i ), ϕ ( t + i ) for all i = 1 , , ..., N, endowed with theuniform norm denoted by (cid:107) · (cid:107) PC ( X ) . Definition 2.1.
A function ω ( · ) ∈ PC ( X ) is called a mild solution for the system (2.2) if it satisfies the following integral-algebraic equation ω ( t ) = h ( t ) , t ∈ [ − r, ,S ( t ) h (0) + (cid:90) t S ( t − s ) (cid:0) B θ u ( s ) + f ( s, ω s , u ( s )) (cid:1) d s, t ∈ [0 , t ] ,G i ( t, ω ( t ) , u ( t )) , t ∈ ( t i , s i ] , i = 1 , , . . . , N,S ( t − s i ) G i ( s i , ω ( s i ) , u ( s i )) + (cid:90) ts i S ( t − s ) B θ u ( s ) d s t ∈ ( s i , t i +1 ] , i = 1 , , . . . , N, + (cid:90) ts i S ( t − s ) f ( s, ω s , u ( s )) d s. (2.3) PPROXIMATE CONTROLLABILITY OF SEMI-LINEAR HEAT EQUATION 5
In this paper, we are interested in showing that the semilinear heat equation withnon instantaneous impulses, memory and delay (1.1) is approximately controllableon [0 , T ]. In this regard, we will assume the following hypothesis for all the remaindof paper: H) M ∈ L ∞ (cid:0) [0 , T ] × [0 , π ] (cid:1) , and the nonlinear functions f , g and G i , are smoothenough so that, for all h ∈ PW and u ∈ L ([0 , T ]; U ) the problem (2.1) admit onlyone mild solution on [ − r, T ].These types of problem have been attracting many researchers. For instant,results in [10, 11, 12, 13] showed the well-posedness of certain classes of nonlinearand noninstantaneous impulsive differential equations. In the next section, we willdiscuss the result for the linear case.3. Approximate Controllability of the Linear Equation
In this section, we shall present some characterization of the approximate con-trollability for a general linear system in Hilbert spaces and prove, for the betterunderstanding of the reader, the approximate controllability of the linear heat equa-tion in any interval [ T − l, T ] using the representation of the semigroup ( S ( t )) t ≥ generated by A, and the fact that φ n ( ξ ) = sin nξ are analytic functions. To thisend, we note that, for all ω ∈ X , ≤ t ≤ T and u ∈ L (0 , T ; U ) the initial valueproblem ω (cid:48) = − Aω + B θ u, ω ∈ X ,ω ( t ) = ω , (3.1)admits only one mild solution given by ω ( t ) = S ( t − t ) ω + (cid:90) tt S ( t − s ) B θ u ( s ) d s, t ∈ [ t , T ] . Definition 3.1. (Approximate Controllability of System (3.1) ) The system (3.1) is said to be approximately controllable on [ t , T ] , if for all ω , ω ∈ X = U = L (0 , π ) , ε > there exists u ∈ L (0 , T ; U ) such that the mild solution ω ( t ) of (3.1) corresponding to u verifies: (cid:13)(cid:13) ω ( T ) − ω (cid:13)(cid:13) X < ε, where (cid:13)(cid:13) ω ( T ) − ω (cid:13)(cid:13) X = (cid:18)(cid:90) π (cid:12)(cid:12) ω ( T, x ) − ω ( x ) (cid:12)(cid:12) d x (cid:19) . Definition 3.2.
For l ∈ [0 , T ) we define the controllability map for the system (3.1) as follows G T l : L ( T − l, T ; U ) −→ X G T l ( v ) = (cid:90) TT − l S ( T − s ) B θ v ( s ) d s. H. LEIVA, W. ZOUHAIR AND M.N. ENTEKHABI
It’s adjoint operator G ∗ T l : X −→ L ( T − l, T ; U ) G ∗ T l ( x ) = B ∗ θ S ∗ ( T − t ) x, t ∈ [ T − l, T ] . Therefore, the Grammian operator Q T l : X −→ X is defined as: Q T l = G T l G ∗ T l = (cid:90) TT − l S ( T − t ) B θ B ∗ θ S ∗ ( T − t ) d t. The following lemma holds in general for a linear bounded operator G : W → Z between Hilbert spaces W and Z. Lemma 3.1. (see [15, 16, 17] ) The equation (3.1) is approximately controllable on [ T − l, T ] if, and only if, one of the following statements holds:a. Rang ( G T l ) = X , b. B ∗ θ S ∗ ( T − t ) x = 0 , t ∈ [ T − l, T ] ⇒ x = 0 , c. (cid:104) Q T l x, x (cid:105) > , x (cid:54) = 0 in X , d. lim α → + α ( αI + Q T l ) − x = 0 , ∀ x ∈ X . Remark 3.1.
The Lemma . implies that for all x ∈ X we have G T l u α = x − α ( αI + Q T l ) − x, where u α = G ∗ T l ( αI + Q T l ) − x, α ∈ (0 , . So, lim α → G T l u α = x, and the error E T l x of this approximation is given by E T l x = α ( αI + Q T l ) − x, α ∈ (0 , , and the family of linear operators Γ αT l : X −→ L ( T − l, T ; U ) , defined for < α ≤ by Γ αT l x = G ∗ T l ( αI + Q T l ) − x, satisfies the following limit lim α → G T l Γ αT l = I, in the strong topology. Lemma 3.2.
The linear heat equation (3.1) is approximately controllable on [ T − l, T ] . Moreover, a sequence of controls steering the system (3.1) from an initialstate y to an ε neighborhood of the final state ω at time T > , is given by (cid:8) u lα (cid:9) <α ≤ ⊂ L ( T − l, T ; U ) , where u lα = G ∗ T l ( αI + Q T l ) − (cid:0) w − S ( l ) y (cid:1) , α ∈ (0 , , and the error of this approximation E α is given by E α = α ( αI + Q T l ) − ( w − S ( l ) y ) , such that the solution y ( t ) = y (cid:0) t, T − l, y , u lα (cid:1) of the initial value problem (cid:26) y (cid:48) = − Ay + B θ u lα ( t ) , y ∈ X , t > ,y ( T − l ) = y , (3.2) satisfies lim α → + y lα (cid:0) T, T − l, y , u lα (cid:1) = ω , PPROXIMATE CONTROLLABILITY OF SEMI-LINEAR HEAT EQUATION 7 that is lim α → + y lα ( T ) = lim α → + (cid:40) S ( l ) y + (cid:90) TT − l S ( T − s ) B θ u lα ( s )d s (cid:41) = ω . Proof.
We shall apply condition (b) from the foregoing Lemma. In fact, It is clearthat S ∗ ( t ) = S ( t ), B ∗ θ = B θ , and we suppose that: B ∗ θ S ∗ ( τ − t ) ξ = 0 , t ∈ [ T − l, T ] . i.e., ∞ (cid:88) n =1 e − n ( T − t ) < ξ, φ n > B θ φ n = 0 , t ∈ [ T − l, T ] . i.e., ∞ (cid:88) n =1 e − n ( T − t ) < ξ, φ n > θ φ n = 0 , t ∈ [ T − l, T ] . i.e., ∞ (cid:88) n =1 e − n ( T − t ) < ξ, φ n > φ n ( x ) = 0 , t ∈ [ T − l, T ] , x ∈ θ. i.e., ∞ (cid:88) n =1 e − n t < ξ, φ n > φ n ( x ) = 0 , t ∈ [0 , l ] , x ∈ θ From Lemma 3.14 from [15], we get that < ξ, φ n > φ n ( x ) = 0 , x ∈ θ Now, since φ n ( x ) = sin( nx ) are analytic functions , we get that < ξ, φ n > φ n ( x ) =0 , ∀ x ∈ [0 , π ] , n = 1 , , . . . ;. This implies that < ξ, φ n > = 0 , n = 1 , , . . . . Since { φ n } is a complete orthonormal set on X , we conclude that ξ = 0. Thiscompletes the proof of the approximate controllability of the linear system (3.1).The remaind of the prove follows from the above characterization of dense rangeoperators. (cid:3) Approximate Controllability of the Semilinear System (1.1)In this section, we shall prove the main result of this paper, the interior approx-imate controllability of the non instantaneous impulsive semi-linear heat equationwith memory and delay (1.1), which is equivalent to prove the approximate con-trollability of the system (2.1).Under hypothesis H, for all h ∈ PW and u ∈ L ([0 , T ]; U ) , the initial value problem ω (cid:48) = − Aω + B θ u + (cid:90) t M ( t, s ) g ( ω s ) ds + f ( t, ω t , u ( t ))d s, ∪ Ni =0 ( s i , t i +1 ) ,ω ( t ) = h ( t ) , [ − r, ,ω ( t ) = G i ( t, ω ( t ) , u ( t )) , ∪ Ni =0 ( t i , s i ] , (4.1) H. LEIVA, W. ZOUHAIR AND M.N. ENTEKHABI admit only one mild solution given by (2.3), and its evaluation in T leads us to thefollowing expression ω ( T ) = S ( T − s N ) G N ( s N , ω ( s N ) , u ( s N )) + (cid:90) Ts N S ( T − s ) (cid:0) B θ u ( s ) + f ( s, ω s , u ( s )) (cid:1) d s = S ( T − s N ) G N ( s N , ω ( s N ) , u ( s N )) + (cid:90) Ts N S ( T − s ) B θ u ( s )d s + (cid:90) Ts N S ( T − s ) (cid:90) s M ( s, m ) g ( ω m )d m + f ( s, ω s , u ( s ))d s. Definition 4.1. (Approximate Controllability of System (2.1) ) The system (2.1) is said to be approximately controllable on [0 , T ] , if for all h ∈ PW and ω ∈ X = U = L (0 , π ) , ε > there exists u ∈ L (0 , T ; U ) such that the mild solution ω ( t ) of (2.1) corresponding to u verifies: (cid:13)(cid:13) ω ( T ) − ω (cid:13)(cid:13) X < ε. Now, we are ready to present and prove the main result of this paper.
Theorem 4.1.
Assume that It exists ρ ∈ C ( R + , R + ) which for all ( t, Φ , u ) ∈ [0 , T ] × PW ( − r, X ) × L ([0 , T ]; U ) , the following inequality holds (cid:13)(cid:13) f ( t, Φ , u ) (cid:13)(cid:13) X ≤ ρ ( (cid:107) Φ( − r ) (cid:107) X ) . (4.2) Then, the non instantaneous impulsive semilinear heat Eq. (1.1) with memory anddelay is approximately controllable on [0 , T ] . Proof.
Given ε > , h ∈ PW and a final state w ∈ X , we want to find a control u lα ∈ L (0 , T ; U ) such that (cid:13)(cid:13) ω lα ( T ) − ω (cid:13)(cid:13) X < ε. We start by considering u ∈ L (0 , T ; U ) and its corresponding mild solution ω ( t ) = ω ( t, , h, u ) , of the initial value problem (4.1), for 0 < α < < l < min { T − s N , r } small enough. We define the control u lα ∈ L (0 , T ; U ) as follow u lα ( t ) = (cid:26) u ( t ) , if 0 ≤ t ≤ T − l,u α ( t ) , if T − l < t ≤ T, (4.3)where u α ( t ) = B ∗ θ S ∗ ( T − t ) ( αI + Q T l ) − (cid:0) ω − S ( l ) ω ( T − l ) (cid:1) , T − l < t ≤ T. (4.4)The corresponding solution ω α,l = ω ( t, s N , G N , u lα ) of the initial value problem(4.1) at time T can be written as follows: PPROXIMATE CONTROLLABILITY OF SEMI-LINEAR HEAT EQUATION 9 ω α,l ( T ) = S ( T − s N ) G N ( s N , ω α,l ( s N ) , u lα ( s N )) + (cid:90) Ts N S ( T − s ) (cid:20) B θ u lα ( s )+ (cid:90) s M ( s, m ) g ( ω α,lm )d m + f ( s, ω α,ls , u lα ( s )) (cid:21) d s = S ( l ) (cid:26) S ( T − s N − l ) G N ( s N , ω α,l ( s N ) , u lα ( s N )) + (cid:90) T − ls N S ( T − s − l ) (cid:20) B θ u lα ( s )+ (cid:90) s M ( s, m ) g ( ω α,lm )d m + f ( s, ω α,ls , u lα ( s )) (cid:21) d s (cid:27) + (cid:90) TT − l S ( T − s ) (cid:20) B θ u α + (cid:90) s M ( s, m ) g ( ω α,lm )d m + f ( s, ω α,ls , u α ( s )) (cid:21) d s, then ω α,l ( T ) = S ( l ) ω ( T − l ) + (cid:90) TT − l S ( T − s ) (cid:20) B ω u α + (cid:90) s M ( s, m ) g ( ω α,lm )d m + f ( s, ω α,ls , u α ( s )) (cid:21) d s. On the other hand, the corresponding solution y lα ( t ) = y ( t, T − l, ω ( T − l ) , u α ) ofthe initial value problem (3.2) at time T is given by: y lα ( T ) = S ( l ) ω ( T − l ) + (cid:90) TT − l S ( T − s ) B ω u α ( s )d s. (4.5)Therefore, ω α,l ( T ) − y lα ( T ) = (cid:90) TT − l S ( T − s ) (cid:90) s M ( s, m ) g ( ω α,lm )d m + f ( s, ω α,ls , u α ( s ))d s, by the hypothesis (4.2) of the theorem we obtain (cid:13)(cid:13) ω α,l ( T ) − y lα ( T ) (cid:13)(cid:13) ≤ (cid:90) TT − l (cid:107) S ( T − s ) (cid:107) (cid:90) s (cid:13)(cid:13) M ( s, m ) g (cid:0) ω α,lm (cid:1)(cid:13)(cid:13) d m d s + (cid:90) TT − l (cid:107) S ( T − s ) (cid:107) ρ (cid:0)(cid:13)(cid:13) ω α,ls ( − r ) (cid:13)(cid:13)(cid:1) ds, since 0 ≤ m ≤ s, < l < r, and T − l ≤ s ≤ T, then m − r ≤ s − r ≤ T − r < T − l, then ω α,l ( m − r ) = ω ( m − r ) and ω α,ls ( − r ) = ω ( s − r ) , (4.6)therefore, for a sufficiently small l we obtain (cid:13)(cid:13) ω α,l ( T ) − y lα ( T ) (cid:13)(cid:13) ≤ (cid:90) TT − l (cid:107) S ( T − s ) (cid:107) (cid:90) s (cid:13)(cid:13) M ( s, m ) g ( ω m ) (cid:13)(cid:13) d m d s + (cid:90) TT − l (cid:107) S ( T − s ) (cid:107) ρ ( (cid:107) ω ( s − r ) (cid:107) ) d s ≤ ε . Hence, by lemma 3.2 we can choose α > (cid:13)(cid:13) ω α,l ( T ) − ω (cid:13)(cid:13) ≤ (cid:13)(cid:13) ω α,l ( T ) − y lα ( T ) (cid:13)(cid:13) + (cid:13)(cid:13) y lα ( T ) − ω (cid:13)(cid:13) < ε ε ε. (cid:3) Remark . In particular of function ρ from condition (4.2) , is ρ ( ξ ) = e ( ξ ) β + η ,with β ≥ . Example
In this section, we’re going to do a numerical validation for the controllabilityresult obtained in lemme 3.2 in which we have shown the approximate controllabilityof the linear heat equation given by (cid:26) y (cid:48) = − Ay + B θ u lα ( t ) , y ∈ X , < t < T,y ( T − l ) = y , (5.1)where the control function is giving by u lα = G ∗ T l ( αI + Q T l ) − (cid:0) w − S ( l ) y (cid:1) , α ∈ (0 , . For the simulations we make the following choices: T = 2 π, l = π , θ =]0 , π [ , alsowe put y = 0 , and for the desired state we take w ( x ) = sin( x ) for all x ∈ [0 , π ] . Desired state w . Now, under the influence of control, we solve the problem (5.1), and we plot thesolution y lα ( T, T − l, y , u α ) for α = , we get PPROXIMATE CONTROLLABILITY OF SEMI-LINEAR HEAT EQUATION 11
Solution of the problem (5.1) for α = . Clearly we can see that this solution takes the same form as the desired state,but the error is still big because α must be tend towards 0 . By decreasing the valueof α , we notice that y lα ( T, T − l, y , u α ) is getting close more and more towards w .Therefore, the result of Lemma 3.2 is hold.6. Final Remark
We strongly believe that the technique adopted here can apply it to prove theinterior controllability of many other partial differential equations.6.1.
Open Problem.
Our first open problem is the semilinear Non-autonomousdifferential equations with non instantaneous impulses, memory and delay ω (cid:48) ( t ) = A ( t ) ω ( t ) + B ( t ) u ( t ) + (cid:90) t M ( t, s ) g ( ω s ) ds + f ( t, ω t , u ( t )) d s, in ∪ Ni =0 ( s i , t i +1 ) ,ω ( s, x ) = h ( s, x ) , in [ − r, ,ω ( t ) = G i ( t, ω ( t ) , u ( t )) , in ∪ Ni =0 ( t i , s i ] , where 0 = s = t < t ≤ s < ... < t N ≤ s N < t N +1 = T are fixed realnumbers, ω ( t ) ∈ R n , u ( t ) ∈ R m , ω t defined as a function from [ − r,
0] to R n by ω t ( s ) = ω ( t + s ), A ( t ) , B ( t ) are continuous matrices of dimension n × n and n × m respectively, the control function u belongs to C (0 , T ; R m ) , h ∈ PW ( − r, R n ) ,f : [0 , T ] × PW (0 , T ; R n ) × R m → R n , g : R n → R n , M ∈ C (0 , T ; R n ) G i :[ t i , s i ] × R n × R m → R n . Open Problem.
Second open problem is about Controllability of non instan-taneous semilinear beam equation with memory and delay z (cid:48)(cid:48) − γ ∆ z (cid:48) + ∆ z = u ( t, x ) + f (cid:16) t, z ( t − r ) , z (cid:48) ( t − r ) , u (cid:17) in ∪ Ni =0 ( s i , t i +1 ) , + (cid:90) t g ( t − s ) h ( z ( s − r, x ))d s,z ( t, x ) = ∆ z ( t, x ) = 0 , on (0 , T ) × ∂ Ω ,z ( s, x ) = ϕ ( s, x ) , in [ − r, × Ω ,z (cid:48) ( s, x ) = ϕ ( s, x ) ,z (cid:48) = ψ i ( t, z ( t ) , z (cid:48) ( t ) , u ( t )) , in ∪ Ni =0 ( t i , s i ] , where Ω is a bounded domain in R N , the damping coefficient γ > , and thereal-valued functions z = z ( t, x ) in [0 , T ] × Ω represents the beam deflection, u in[0 , T ] × Ω is the distributed control, g acts as convolution kernel with respect to thetime variable.6.3. Open Problem.
Another example where this technique may be applied isthe strongly damped wave equation with Dirichlet boundary conditions with noninstantaneous impulses, memory and delay in [0 , T ] × Ω , y (cid:48)(cid:48) + β ( − ∆) / y (cid:48) + γ ( − ∆) y = θ u + (cid:90) t h ( s, y ( s − r ) , u ( s ))d s, in ∪ Ni =0 ( s i , t i +1 ) ,y = 0 , on ∂ Ω ,y ( s ) = φ ( s ) , in [ − r, ,y (cid:48) ( s ) = φ ( s ) , in [ − r, ,y (cid:48) ( t ) = g i ( t, y ( t ) , y (cid:48) ( t ) , u ( t )) , in ∪ Ni =0 ( t i , s i ] . In the space Z / = D (( − ∆) / ) × L (Ω) , where Ω is a bounded domain in R N ,θ is an open nonempty subset of Ω , θ denotes the characteristic function of theset θ, the distributed control u ∈ L (0 , T ; L (Ω)) . φ , φ are piecewise continuousfunctions. References [1] Liu, Xinzhi. ”Practical stabilization of control systems with impulse effects.” Journal of math-ematical analysis and applications 166.2 (1992): 563-576.[2] Mcrae, Farzana A. ”Practical stability of impulsive control systems.” Journal of mathematicalanalysis and applications 181.3 (1994): 656-672.[3] Qin, Shulin, and Gengsheng Wang. ”Controllability of impulse controlled systems of heatequations coupled by constant matrices.” Journal of Differential Equations 263.10 (2017):6456-6493.[4] Phung, Kim Dang, Gengsheng Wang, and Yashan Xu. ”Impulse output rapid stabilizationfor heat equations.” Journal of Differential Equations 263.8 (2017): 5012-5041.[5] Guevara, Cristi, and Hugo Leiva. ”Controllability of the impulsive semilinear heat equationwith memory and delay.” Journal of Dynamical and Control Systems 24.1 (2018): 1-11.[6] Duquea, Cosme, et al. ”Approximate controllability of semilinear strongly damped waveequation with impulses, delays, and nonlocal condi-tions.”
PPROXIMATE CONTROLLABILITY OF SEMI-LINEAR HEAT EQUATION 13 [7] Ahmed, Hamdy M., et al. ”Approximate controllability of noninstantaneous impulsive Hilferfractional integrodifferential equations with fractional Brownian motion.” Boundary ValueProblems 2020.1 (2020): 1-25.[8] Kumar, Vipin, and Muslim Malik. ”Controllability results of fractional integro-differentialequation with non-instantaneous impulses on time scales.” IMA Journal of MathematicalControl and Information (2020).[9] Agarwal, Ravi, D. Oregan, and S. Hristova. ”Stability by Lyapunov like functions of nonlineardifferential equations with non-instantaneous impulses.” Journal of Applied Mathematics andComputing 53.1-2 (2017): 147-168.[10] Pierri, Michelle, Donal ORegan, and Vanessa Rolnik. ”Existence of solutions for semi-linearabstract differential equations with not instantaneous impulses.” Applied mathematics andcomputation 219.12 (2013): 6743-6749.[11] Pierri, Michelle, Hernn R. Henrquez, and Andra Prokopczyk. ”Global solutions for abstractdifferential equations with non-instantaneous impulses.” Mediterranean Journal of Mathe-matics 13.4 (2016): 1685-1708.[12] Anguraj, A., and S. Kanjanadevi. ”Existence of mild solutions of abstract fractional differ-ential equations with non-instantaneous impulsive conditions.” J. Stat. Sci. Appl 4 (2016):01-02.[13] Hernndez, Eduardo, and Donal ORegan. ”On a new class of abstract impulsive differentialequations.” Proceedings of the American Mathematical Society 141.5 (2013): 1641-1649.[14] Malik, Muslim, et al. ”Controllability of non-autonomous nonlinear differential system withnon-instantaneous impulses.” Revista de la Real Academia de Ciencias Exactas, Fsicas yNaturales. Serie A. Matemticas 113.1 (2019): 103-118.[15] Curtain, Ruth F., Pritchard, A.J., Infinite Dimensional Linear Systems. Lecture Notes inControl and Information Sciences, 8. Springer Verlag,Berlin (1978).[16] Curtain, Ruth F., and Hans Zwart. An introduction to infinite-dimensional linear systemstheory. Vol. 21. Springer Science Business Media, 2012.[17] Leiva, Hugo, N. Merentes, and J. Sanchez. ”A characterization of semilinear dense rangeoperators and applications.” Abstract and Applied Analysis. Vol. 2013. Hindawi, 2013. School of Mathematical Sciences and Information Technology,, Universidad YachayTech,, San Miguel de Urcuqui, Ecuador
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