9) Consider the following optimal control problem. The system's dynamic equation is

i(t) = Ax(t) + bu(t),

x(0) = (50)¹,

where r(t) = R², u(t) = R, and the matrices A and b are

0

A =

^-(+).

00

Minimize the cost functional J, defined as

where

July 29, 2023

00

Compute the gain matrix K € R¹x2 such that

according to the optimal control.

J = = 1/2 (x(1)¹ Qa(t) + u(t)²) dt,

b =

10

Q=

e-(69)

01

-(9).

u(t) =−Kr(t)

DRAFT

210