]> 8.8 Confirmation of the results by the Wiener – Hopf method

8.8 Confirmation of the results by the Wiener – Hopf method

Our objective is to justify and confirm the results obtained via the formal spectral–factorization method. To do this, an approach similar to that of [19] will be used. First of all, we show that there is a unique u L2(0,) minimizing (8.2). Next, using the Wiener–Hopf technique, the Laplace transform û H2(Π+) of u will be found. Finally, the formula obtained for û will be compared with an expression for û arising from (8.22).

Notice that the output function y of system (8.5) can be represented as

y = yh + yn ,

where

yh(t) = c0Akϕ(t kr r),t [kr, (k + 1)r] (8.34)

denotes the homogeneous part of the output corresponding to the response to a nonzero initial condition ϕ on [r, 0] and u = 0, and where

yn(t) = k=0c 0Akb 0(T[(k + 1)r]u)(t) (8.35)

denotes the nonhomogeneous part of the output corresponding to response to ϕ = 0 on [r, 0] and a nonzero forcing term u. Here {T(t)}t0 is the semigroup of right shifts on L2(0,).

Now,

yh L2(0,)2 = k=0krkr+r c 0Akϕ(t kr r) 2dt k=0(A)kc 0 22 ϕ L2(r,0;2)2 .

The spectrum of the matrix A lies in the open unit disk, and thus yh L2(0,) for any initial condition ϕ L2(r, 0; 2).

Similar arguments lead, through the estimate

yn L2(0,) k=0c 0Akb 0 uL2(0,) ,

to the conclusion that the mapping : L2(0,) uy n L2(0,) is linear and bounded. Thus (8.2) can be rewritten as

J = yh + uL2(0,)2 + u L2(0,)2 (8.36)

and, by Curtain and Pritchard [15, Theorem 12.1, p. 254], J achieves its minimal value at u L2(0,) defined as the unique solution of the equation

( + I)u = y h (8.37)

A direct solution of (8.37) is complicated, to determine the optimal control function u L2(0,), the Wiener–Hopf technique will be useful. By Parseval’s theorem, instead of minimizing (8.36) one can minimize

J = 1 2πĝ(jω)û(jω) + ŷ h(jω) 2 + û(jω) 2 dω (8.38)

where ĝ H(Π+) is given by

ĝ(s) = esrc 0(I esrA)1b 0 = aesr 1 + be2sr (8.39)

and where û, ŷh H2(Π+), with ŷh given by

ŷh(s) = aesr 1 + be2sr besrϕ̂ 1F (s) + ϕ̂2F (s) (8.40)

are the Laplace transform of g, u, and yh respectively, and ϕ̂1F (s) and ϕ̂2F(s) are the finite Laplace transforms of ϕ1 and ϕ2, the components of ϕ. Under the Laplace transform, the operator is similar to the multiplication operator H2(Π+) ûĝû H2(Π+) with ĝ H(Π+) given by (8.39). Observe that (8.39) agrees with (8.11).

Applying once more the result from [15, Theorem 12.1, p. 254] we establish that J achieves its minimum at û L2(j) (on a larger space), given by

û(jω) = ŷh(jω)ĝ(jω) 1 + ĝ(jω) 2 ,ω .

Hence, by the orthogonal projection theorem,

ûopt(s) = P+ jω ŷh(jω)ĝ(jω) 1 + ĝ(jω) 2 (s) ,

where P+ denotes the orthogonal projector from L2(j) onto H2(Π+), which – by the uniqueness theorem for harmonic functions – may be treated as a closed subspace of L2(j). The spectral–factorization of Section 8.4 will be used to determine ûopt. Notice that ϑ̂, 1ϑ̂ H(Π+), where ϑ̂ is the spectral factor defined by (8.13) (with p and q given by (8.14)). Now

ûopt(s) = 1 ϑ̂(s)P+ jω ŷh(jω)ĝ(jω) ϑ̂(jω) (s) .

Taking into account the formulae (8.13), (8.39) and (8.40), we get

ŷh(s)ĝ(s) ϑ̂(s) =(1 p2) qesr 1 + be2sr ϕ̂1F (s) + esrϕ̂ 2F (s) + + (1 p2) 1 p + qe2sr ϕ̂2F (s) + q2esrϕ̂ 1F (s) (8.41)

By replacing in (8.40) the initial vector ϕ1 ϕ2 by

(1 p2)q a 1 bϕ2 ϕ1 ,

we establish that the first term in the right–hand side of (8.41) is the Laplace transform of a function which belongs to L2(0,), or equivalently that the expression represents a function in H2(Π+). The second term in the right–hand side of (8.41) has poles in Π+ and therefore does not belong to H2(Π+). Hence

ûopt(s) = 1 ϑ̂(s)(1 p2) qesr 1 + be2sr ϕ̂1F (s) + esrϕ̂ 2F (s) = = (1 p2) qesr p + qe2sr ϕ̂1F (s) + esrϕ̂ 2F (s) (8.42)

According to the results presented in Section 8.6, (8.42) should agree with

û(s) = g#(sI 𝒜 c)1ϕ (8.43)

From (8.19) and (8.20), we find

g#x =(1 p)x 2(0) + (b q)x1(r) = (1 p)[x2(0) qx1(r)] = =(1 p)[x2(0) + px2(0)] = (1 p2)x 2(0)x D(𝒜c) (8.44)

Thus, to evaluate the right–hand side of (8.43), complete knowledge of (sI 𝒜c)1ϕ is not required: it is enough to observe that

((sI𝒜c)1ϕ)(0) = esr p + qe2sr qesr p q qesr ϕ̂1F (s) ϕ̂2F (s) (8.45)

On substituting (8.45) and (8.44) in (8.43), we get (8.42), which confirms the validity of the results obtained via the formal spectral–factorization method. Correctness of the expression (8.44) for the optimal feedback implies, via the closed–loop Lyapunov equation (8.28), the correctness of the optimal cost expression obtained in Section 8.7.

Remark 8.8.1. The operators:

H2(Π+) ŷ hP+ jωŷh(jω)ĝ(jω) 1 + ĝ(jω) 2 H2(Π+) H2(Π+) ŷ hP+ jωŷh(jω)ĝ(jω) ϑ̂(jω) H2(Π+)

are called the Toeplitz operators with symbols jω ĝ(jω) 1 + ĝ(jω) 2 L(j) and jω ĝ(jω) ϑ̂(jω) L(j), respectively.