This document discusses constrained optimization problems in control systems. It introduces three types of constraints: 1) point constraints 2) differential equation constraints and 3) isoperimetric constraints. The Lagrange multiplier method is used to extend the necessary conditions for unconstrained problems to problems with constraints. Constrained problems are transformed into an augmented function involving Lagrange multipliers to determine the constrained extrema. Examples demonstrate applying this method to problems with different constraint types.

1. OPTIMAL and MULTIVARIABLE
CONTROLS
LECTURE # 10
CONSTRAINED EXTREMA

2. Constrained Extrema
• The discussion so far has been about
functionals involving assuming that the
components of x are independent.
• In control problems the trajectory is
determined by the control u
– We wish to consider functionals of n + m functions
– Only m are independent known as controls
• Need to extend the necessary conditions to
include problems with constraints.

3. Constrained Minimization of
Functions

4. The Elimination Method
• Function to be considered

5. The Elimination Method
• y1 and y2 are independent, then ∆y1 and ∆y2
could be selected arbitrarily.
• y1 and y2 are constrained to lie on a specified
line, so ∆y1 and ∆y2 are not independent.
• Solving

6. The Lagrange Multiplier Method
• Consider the augmented function
• With p a variable (the Lagrange multiplier) whose value is yet
to be determined.
• For values of y1 and y2 that satisfy the constraining relation,
augmented function
– fa = f (regardless of the value of p)
• By satisfying the constraint and minimizing fa, the constrained
extreme point can be found.

8. The Lagrange Multiplier Method
• c
• For the coefficients of ∆p
• The remaining two terms must add up to zero.
• Since ∆y1 and ∆y2 are not independent, so to make it zero “p”
is selected such that the corresponding coefficients are zero.
• Thus,
• Solving these equations give

9. The General Problem

10. The General Problem

11. General Method Using Lagrange
Multipliers

12. General Method Using Lagrange
Multipliers
•
•
•

13. Example 4.5-2

14. Example 4.5-2

17. Constrained Minimization of
Functionals
• The problem of solving functionals in the
presence of following constraints is
considered
– Point constraints.
– Differential equation constraints.
– Isoparametric constraints

18. 1. Point Constraints

19. Point Constraints
• Solution via Lagrange multipliers
– The first step is to form the augmented
functionals by adjoining the constraining
relations to J, which yields

23. Example 4.5-3

24. Example 4.5-3

25. 2. Differential Equation Constraints

26. Differential Equation Constraints
• Solution via Lagrange multipliers

29. Differential Equation Constraints
• Since, the constraints are satisfied, we can choose the n
Lagrange multipliers arbitrarily.
• Select p’s so the coefficients of n of the components of ᵟw(t)
are zero in the interval [t0,tf]
• The remaining (n+m)-n = m components of ᵟw(t) are then
independent, thus, the coefficients must be zero.
• Hence, in addition to the constraint equations following
equations should also be satisfied

30. Differential Equation Constraints

31. Example 4.5-4

32. Example 4.5-4

33. Example 4.5-5

37. 3. Isoperimetric Constraints
• To find the curve having a fixed length enclosing the
maximum area.
• The problems having any constraints of the form are
termed as isoperimetric constraints.
• The ci’s are specified constraints.
• In control problem such constraints enter in the form
of total fuel or energy available to perform a required
task.

38. Isoperimetric Constraints

40. Isoperimetric Constraints

41. Example 4.5-6

42. Example 4.5-6

43. Example 4.5-7

44. Example 4.5-7

46. Summary
• This means to determine the necessary conditions for an
extremal we form ga () and write the corresponding Euler eqs.
• The constraining relations should also be satisfied