Consider the time-varying continuous-time system i (t) = A(t)x(t) + B(t)u(t), x(to) = To E Rn (1) where A : R - Rnxn, B : R - R"XP and u : R - RP are continuous functions. Also, denote the transition matrix of the system by DA(t, to). Note that whenever needed, we consider the function u on the interval [to, ti] to get u : [to, ti] - RP. Denote by C the set of all continuous functions from [to, ti] to RP. Then C is a vector space and the inner product in C is defined as (u, v)e = u(t) Tv(t ) dt As a result, the induced norm by this inner product is defined as llulle = (u, u)e Now, for a given u E E, define L(u) = $4(t1, T) B(T)u(T ) dT Note that L(u) E R". It can be easily verified that the mapping L : C - R" is a linear operator between vector spaces e and Rn. . The solution for system (1) is x(t) = bA(t, to)xo + $A(t, T) B(T) u(T ) dr . System (1) is uniformly stable if and only if there exists y > 0 such that 1 5 211 ( 07 ' 7 ) Vol1 : 07 ZIA . System (1) is uniformly exponentially stable if and only if there exist y, a > 0 such that ( 07-7)0_ 2 1 5 2 11 ( 07 7 ) Vol1 : 07 ZZA . System (1) is called controllable on [to, ti] if and only if for every given To, $1 E R", there exists an input u(t) for t E [to, ti] such that if x(to) = To, then x(t1) = $1. . System (1) is called controllable to zero on [to, ti] if and only if for every given To E R", there exists an input u(t) for t E [to, ti] such that if x(to) = To, then x(t1) = 0. . System (1) is controllable on [to, ti] if and only if L is surjective, that is, R(L) = R". . Denote by L*, the adjoint operator corresponding to L. Let We(to, t1) = LL* and note that We(to, t1) is a matrix. We(to, 1) is called controllability Gramian. . Consider system (1) and assume A = A(t) and B = B(t) are time invariant and p = 1. Then, the system is controllable if and only if rank(C) = n where C = [B AB ... An ' B ] The matrix C is called controllability matrix and R(C) is called controllabe subspace. (8 points) Suppose system (1) is controllable on [to, ti]. Prove that it is controllable on [to, t2], for all t2 2 t1