On the approximate controllability of semilinear control systems

ABOUT THE AUTHORS Divya Ahluwalia received her PhD degree in Mathematics from Indian Institute of Technology, Roorkee (India) in 2004. Currently, she is an associate professor in the University of Petroleum and Energy Studies, Dehradun, India. Her research area includes controllability of deterministic systems. N. Sukavanam received his BSc (Maths) degree from University of Madras, India in 1977 and MSc (Maths) from the same university in 1979 and PhD (Maths) from IISC Bangalore, India in 1985. At present, he is working as a professor in the Department of Mathematics IIT Roorkee (India). His research interests include nonlinear analysis, control theory and robotics and control. Anurag Shukla received his MSc (Maths) degree from IIT Roorkee, India in 2011. He received his PHD degree from IIT Roorkee, India in 2016. At present, he is working as an assistant professor in the University of Petroleum and Energy Studies, Dehradun. His research interests are nonlinear analysis, control theory and optimal controls. PUBLIC INTEREST STATEMENT Controllability is an important concept pertaining to any control system. It determines whether the state of the system can be steered to a given target state in a prescribed time interval or not. Therefore, it plays a very important role in the analysis and design of control systems. Also, the noise or stochastic perturbation is omnipresent and unavoidable in nature as well as in man-made systems. So, we have to move from deterministic systems to stochastic systems. Many practical problems contain a delay term in their respective control equations. Therefore, in this paper, we discuss the approximate controllability of semilinear stochastic systems with multiple delays in control using fixed point theorem technique. Received: 01 August 2016 Accepted: 16 November 2016 First Published: 01 December 2016


PUBLIC INTEREST STATEMENT
Controllability is an important concept pertaining to any control system. It determines whether the state of the system can be steered to a given target state in a prescribed time interval or not. Therefore, it plays a very important role in the analysis and design of control systems. Also, the noise or stochastic perturbation is omnipresent and unavoidable in nature as well as in man-made systems. So, we have to move from deterministic systems to stochastic systems. Many practical problems contain a delay term in their respective control equations. Therefore, in this paper, we discuss the approximate controllability of semilinear stochastic systems with multiple delays in control using fixed point theorem technique. Shukla, Sukavanam, and Pandey (2015a, 2015b studies the controllability of semilinear system with delay in state and control using fixed point theorems in 2015-2016. The controllability theory for abstract linear control systems is almost complete. Sukavanam and Tafesse (2011) obtained some sufficient conditions for approximate controllability of semilinear delay system using fixed point approaches in 2011. The necessary and sufficient conditions for various types of controllability for abstract linear equations have been considered in George (1995). One of the principal results on approximate controllability is that the linear control system (*) is approximate controllable on [0.T] if and only if "[S(t)B]*h = 0" for 0 ≤ t ≤ T implies h = 0 in V, where * denotes the adjoint of an operator (see Balakrishnan, 1976;Fattorini, 1967). Fattorini (1967) has given exact controllability results for linear parabolic equations. Triggiani (1975), Russell (1978) and Lions (1988) have proved some exact and approximate controllability results for linear control systems. Recently authors Russell (1978), Sakthivel, Ganesh, and Anthoni (2013), Seidman and Zhou (1982), Shukla, Arora, and Sukavanam (2015), Shukla, Sukavanam, Pandey, and Arora (2016) established some sufficient conditions for controllability of semilinear systems of integer and fractional order systems using Fixed Point theorems. But, in the case of an abstract semilinear control systems, the controllability has been shown under various complex and restricted conditions in literature. Approximate controllability of abstract semilinear systems of the form (1.1) has been studied by Zhou (1983Zhou ( , 1984. Naito (1987) has given simpler conditions for approximate controllability of the system of the form (1.1). The conditions of Naito (1987) are as follows: (a) For every p ∈ Z there exists a q ∈ R(B) such that Lp = Lq where L is the operator defined as in (2.2) and R(B) denotes the closure of R(B).
In Mahmudov, Vijayakumar, Ravichandran, and Murugesu (2015), Mahmudov et al. (2016) the uniform boundedness condition (d) was replaced by the growth condition for F, namely, (e) ||Fx|| Z ≤ a||x|| Z + b, where a ≥ 0 and b is any nonnegative constant such that MTa(1 + c) < 1; the constants T and c being system constants and M is such that ||S(t)|| ≤ M for t ∈ [0, T]. (1.2) In Naito (1987), the approximate and exact controllability of the restricted system (1.1) with B = I, the identity operator is proved for any nonlinear function f satisfying condition (c) of which the heat equation considered by Liu and Williams (1997), Klamka (2000) is a particular case. In George (1995), approximate controllability of semilinear system (1.1) with A = A(t) has been proved under certain conditions which are as follows: (f) f(t, x) satisfies the monotone condition given by and the assumptions (a), (b), (e) hold with In this paper, we will prove the approximate controllability of the semilinear control system (1.1) for an extended class of nonlinear functions f(t, x) satisfying the monotonicity condition. Thus, it is proved that we no more require the complex inequalities such as (i) to prove the approximate controllability of such systems. Let K be an operator from Z into itself defined as:

Notations
and L and N be operators from Z into V defined as: It is evident that hypothesis (a) is equivalent to the condition Z = N 0 (L) + R(B). Moreover, Z can be decomposed as Z = N 0 (L) + N ⟂ 0 (L). Also, under hypothesis (a) we can define a map P: as follows: The operator P is well-defined, linear and continuous (see Mahmudov et al., 2016, Lemma 1). From continuity of P it follows that ||Pu|| Z ≤ c||u|| Z , for some constant c ≥ 0.

Controllability results
In this section, we will discuss the approximate controllability of the deterministic semilinear control system (1.1) under the following two cases: (i) When B = I, the identity operator, and Before proving the main theorems, we prove some lemmas.
, then z can be uniquely decomposed as z = n + q:n ∈ N 0 (L); q ∈ R(B) where q = Pu and n = n 1 − n 0 .
Lemma 2 Under the condition ( j) the operator K:Z → Z satisfies the condition Since S(t) is a strongly continuous semigroup, we have f(t) ∈ D(A) and (3.1) or ‖u‖ ≤ ‖z‖.
Proof Let x u (t) be a mild solution of (3.8) corresponding to a control u and consider the following system The mild solution of (3.8) and (3.9), respectively, can be written as: Subtracting (3.11) from (3.10), we get,

or x u − y v = KNx u − KNy v
Taking inner product on both sides with Nx u − Ny v , we get, (3.6) and (3.9) y v (0) = 0. (3.10) Now the LHS satisfies condition (iv) and therefore it is less than or equal to −β||x u − y v || 2 and from Lemma 2 RHS is nonnegative. This is possible only when x u = y v in Z which implies that F(x u ) = F(y v ) where F is the Nemytskii operator defined by f. This in turn implies from (3.12) that x u (t) = y v (t) for all t ∈ [0, T]. Therefore, the set of all solutions of the linear system is equal to the set of all solutions of the semilinear systems i.e. R T (F) ⊃ R T (0) which is dense in V and this implies the approximate controllability of the system (3.7).
Remark When B = I, the linear system (3.8) is always approximately controllable because for any given initial value x u (0) and the final value x u (T) of the state variable x u (t) we can always find a control u ∈ Y given by u(t) = [x u (T) + tAx u (T)]/T. Hence, in this case the system (3.7) is approximately controllable only under the conditions (ii), (iii) and (iv). In George (1995) the approximate controllability is proved under the conditions given in Section 1. Now we no more require the inequality condition (i) to prove the approximate controllability of the semilinear control systems.

Theorem 2 The semilinear system (1.1) is approximately controllable under the assumptions (a), (b), (f) and (k) R(F) ⊆ R(B)
Proof Let y v (t) be a mild solution of the linear system (*) corresponding to a control v which can be written as y v (t) = ∫ t 0 S(t − s)Bv(s)ds. Then, it follows that the system (1.1) will have a unique mild solution for a given control u as f(t, x) satisfies the monotone condition (f).
Let y v (t) be a mild solution of (1.1) with control v − w. Then is also a solution of (1.1) with control v + n.
If Fx u ∈ R(B) then Fx u = Bw 1 (t) for some w 1 ∈ Y, since R(F) ⊆ R(B) (given). Also, for a given ε > 0 there exists a w 1 in Y such that ||F(x u ) − Bw 1 (t)|| Z ≤ ɛ. Now let z w (t) be a mild solution of (1.1) corresponding to a control v + n. Then From (3.13) we get, Hence the result.
Proof Consider the dynamical system dm n (t) dt = Am n (t) + n(t):m n (0) = 0. It has the unique mild solution m = Kn ∈ M 0 . Also since a ∈ Z there exists a unique q ∈ R(B) and n ∈ N 0 (L) as in (Lemma 1) such that Now q ∈ R(B) implies that for any given ε > 0 there exists a w ∈ Y such that ||q − Bw(t)|| < ɛ. Define the function g 1 (t, x) = g(t, x + m(t)) + q(t). It can be easily seen that g 1 (t, x) satisfies the monotone condition (f). Let G denotes the Nemytskii operator corresponding to g 1 defined as [Gx](t) = g 1 (t, x(t)). As R(g) H ⊆ R(B) in V and q ∈ R(B) in Z it can be easily seen that R(G) ⊆ R(B) in Z and hence for every x ∈ Z and δ > 0 there exists a w 1 ∈ Y such that ||Gx − Bw 1 || ≤ δ. Now consider the pair of systems as follows: Using Theorem 2 it can be shown that the mild solution of y v (t) of the first equation and the mild solution x u (t) of (*) can be chosen such that ||y v (T) − x u (T)|| ≤ ε 1 for any ε 1 > 0, by choosing δ depending on ε 1 and the control v as v = u − w 1 . By adding (3.15) and (3.16) and using (3.14) and (ℓ) it can be shown that y v (t) + m n (t) is the mild solution of (1.1). Since m n (T) = 0 it follows that approximate controllability of (*) implied that of (1.1).

Examples
Example 1 Let V = L 2 (0, π) and A = −d 2 /dx 2 with D(A) consisting of all y ∈ V with d 2 y/dx 2 ∈ V and y(0) = 0 = y(π). Put n (x) = (2∕ ) 1∕2 sin(nx), 0 ≤ x ≤ , n = 1, 2, … , then { n : n = 1, 2, …} is an orthonormal basis for V and φ n is an eigenfunction corresponding to the eigenvalue n = −n 2 of the operator −A, n = 1,2, …. Then the C 0 -semigroup S(t) generated by −A has e n t as the eigenvalues and φ n as their corresponding eigenfunctions (Mahmudov et al., 2016). Now define an infinite dimensional space  (2001), Naito has shown the approximate controllability of the heat control (3.14) a = q + n f (t, y) = z(t) + g(t, y) where g(t, y) = 2b‖y‖ V 1 + b‖y‖ V 2 + a system by assuming uniform boundedness of f(t, y) in the space V. In Mahmudov et al. (2015), approximate controllability was shown under some restrictions on the constants b and T. Now, the approximate controllability of the system follows without any conditions on b and T.
Example 2 Let Ω be a bounded open set in R n with smooth boundary ∂Ω = Γ. Consider the following distributed parameter system where p:[0, T] × Ω → R is such that p(t, z) ≥ c > 0, for some constant c, and is Lipschitz w.r.t. the t variable, C 1 in the z variable and t → ‖p(t, ⋅)‖ ∞ ∈ L ∞ + . Assume that g:[0, T] × Ω × R → R is a nonlinear function such that it is measurable in (t, z) and continuous in x and |g(t, z, The assumption on p(⋅, ⋅) imply that ‖A(t)x − A(s)x‖ ≤ k�t − s� ‖x‖ for some constant k > 0. By Poincare's inequality, there exists μ > 0 such that For x ∈ D, ||A(t)x|| = sup{(A(t)x, v):‖v‖ V < 1} ≤ ‖p(t)‖ C 1 (Ω) ‖x‖ H 2 (Ω) (by Cauchy and Poincare's inequalities). Define f:[0, T] × V → V by Thus f satisfies condition (e) with a = ‖b(z)‖ L 2 [Ω] . If −g(t, z, x) is monotonically increasing with respect to x it follows that f satisfies Now denoting x(t) = x(t, ⋅) ∈ L 2 [Ω] and u(t) = u(t, ⋅) ∈ L 2 [Ω] the system (4.3) takes the form In George (1995), it is shown that the system (4.33) is approximately controllable if a is sufficiently small and the system satisfies the conditions.