(b) Derive the conditions for the existence of singular intervals in linear Time ()pti- (5) mal
Problems.

6. (a) The system . (5) il(t) = X2(t) i2(t) = —C2(t) + u(t)
is to be controlled to minimize the performance measure

2 ۰
where the constraining equations c(to) is specified, the final state (ലട free and 1] is
given. The admissible control values are constrained by
—1l u(t) 1 for t € (to, tf] —2 T2(t) 2 forte
{to, t/

Find the necessary conditions for optimal control.

(b) What do you mean by minimum energy problem? Explain with the help of an (4) example.
How will you find necessary conditions for optimal control for this problem?

PART C
7. (a) A first order discrete time system is described by the difference equation (9)
(k + 1) = —0.5r(k) + u(k)

The performance measure to be minimized is 2

‎(൪)‏ دل

‎k=0

‎and the admissible states and controls are constrained by
-0.2 x(k) 0.2. ಹೊಡಿ

‎-0.1 u(k) 0.1. k

‎Carry out the computational steps required to determine the optimal control
iaw by using dynamic programming. Quantize both u(k) and x(k) in steps of
0.1 about zero and use linear interpolation. What is the optimal control
sequence for an initial taste value of 0.2.

‎(b) State and explain Principle of Optimality. (3)

‎8. (a) What is the use of Hamilton Jacobi Bellman(HJB) in optimal control theory? (3)
(b) Given a first order system (9)

‎2(0) =_2x(t) + u(t) with

‎perf ormance measure

‎J = 52%(t)) +5 [ ‏2ی‎