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Then the system is controllable if and is complex, then its conjugate λ only if there exists a feedback control u(t) = Kx(t) for some feedback matrix K such that the spectrum of A + BK is equal to Σ. Proof. We start by noting two facts: (i) Controllability is invariant under non-singular linear transformations of state space. In case m = 1, that is, we are dealing with the system: dx (t) = Ax(t) + bu(t), dt u(t) ∈ R and b ∈ Rn , then assuming (A, b) is a controllable pair, there exists a non-singular linear transformation T : Rn → Rn such that the above system transforms to ⎡ ⎤ ⎡ ⎤ 0 0 1 0 ...

7) and hence we have proved the following theorem. 2. 1) is controllable in [0, T ]. (ii) W (0, T ) > 0 (positive-definite). . AB .. An−1 B = n. When the system is controllable, we shall refer to (A, B) as a controllable pair. It is of some interest to establish that if the system is controllable, then we can transfer the state x0 at time t0 to the state x1 at time t1 by using a control u∗ , which has the least energy t1 t0 |u∗ (s)|2 ds 2 Controllability, observability, stabilizability, and detectability 17 among all controls that transfer the phase (x0 , t0 ) to the phase (x1 , t1 ).

1 Introduction In many control problems the quadratic criterion is not the most appropriate. We consider a disturbance attenuation problem when the full state vector can be measured and introduce the so-called H ∞ -optimal control problem. There is now a vast literature on this topic. For two textbook presentations, the reader is referred to T. Bas¸ar and P. Bernhard [1] and B. A. Francis [1]. 1) ⎩ z(t) = Cx(t) + Du(t), where u is interpreted as the controls and z as controlled outputs that are to be made small in the presence of exogenous disturbances w.