最优控制理论 第三章

where En (a row vector).

Lagrangian function L: En x Em x En x Eq x E1 E1 is
where Eq is a row vector, whose components are called Lagrange multipliers. Lagrange multipliers satisfy the complimentary slackness conditions:
Remark 3.1: We should have H=0F+f in (3.7) with 0 0. However, we can seBiblioteka Baidu 0=1 in most applications
Remark 3.2: If the set Y in (3.6) consists of a single point Y={k}, then as in (2.75), the transversality condition reduces to simply *(T) equals to a constant to be determined, since x*(T)=k. In this case, salvage value function S can be disregarded.
Example 3.1: Consider the problem:
subject to
Note that constraints (3.16) are of the mixed type (3.3). They can also be rewritten as 0 u x. Solution: The Hamiltonian is
The function is quasiconcave if (3.23) is relaxed to
is strictly concave if y z and p (0,1), and (3.23)
holds with a strict inequality. is convex, quasiconvex, or strictly convex if - is concave, quasiconcave, or strictly concave, respectively.
Theorem 3.1
Let (x*,u*,,μ,,) satisfy the necessary conditions in (3.11). If H(x,u,(t),t) is concave in (x,u) at each t[0,T], S in (3.2) is concave in x, g in (3.3) is quasiconcave in (x,u), a in (3.4) is quasiconcave in x, and b in (3.5) is linear in x, then (x*,u*) is optimal. The concavity of the Hamiltonian with respect to (x,u) is a crucial condition in Theorem 3.1. So we replace the concavity requirement on the Hamiltonian in Theorem 3.1 by a concavity requirement on H0, where
Also note that the optimal control must satisfy
and 1 and 2 must satisfy the complementary slackness conditions
It is obvious for this simple problem that u*(t)=x(t) should be the optimal control for all t[0,1]. We now show that this control satisfies all the conditions of the Lagrangian form of the maximum principle.
Chapter 3 The Maximum Principle: Mixed Inequality Constraints
Mixed inequality constraints: Inequality constraints involving control and possibly state variables. Examples:
Theorem 3.2
Theorem 3.1 remains valid if
and, if in addition, we drop the quasiconcavity requirement on g and replace the concavity requirement on H in Theorem 3.1 by the following assumption: For each t[0,T], if we define A1(t) = {x|u, g(x,u,t) 0 for some u}, then H0(x,(t),t) is concave on A1(t), if A1(t) is convex.If A1(t) is not convex,we assume that H0 has a concave extension to co(A1(t)), the convex hull of A1(t).

Since the right-hand side of (3.22) is always positive, u*= x satisfies (3.17). Note that 2 = e1-t 0 and x-u* = 0, so (3.21) holds.
3.2 Sufficiency Conditions Let D En be a convex set. A function : D E1 is concave, if for all y,z D and for all p[0,1],
g(u,t) 0 , g(x,u,t) 0 .
3.1 A Maximum Principle for Problems with Mixed Inequality Constraints

State equation:
where x(t) En, u(t) Em and f: En x Em xE1 En is assumed to be continuously differentiable. Objective function:


The adjoint vector satisfies the differential equation
with the boundary conditions
where Ela and Elb are constant vectors. The necessary conditions for u* by the maximum principle are that there exist , , , such that (3.11) holds, i.e.,
Elb
are
Interesting case of the terminal inequality constraint:
where Y is a convex set, X is the reachable set from the initial state x0, i.e.,
Notes: (i) (3.6) does not depend explicitly on T. (ii) Feasible set defined by (3.4) and (3.5) need not be convex. (iii) (3.6) may not be expressible by a simple set of inequalities.
Furthermore, if the terminal time T in (3.1)-(3.5) is unspecified, there is an additional necessary transversality condition for T* to be optimal
if T* (0,) .
subject to (3.1) and (3.3)-(3.5).
The standard Hamiltonian is
where F: En x Em x E1 E1, and S: En x E1 E1 are continuously differentiable and T is the terminal time.

u(t), t[0,T] is admissible if it is piecewise continuous and satisfies the mixed constraints.
3.3 Current-Value Formulation
Assume a constant continuous discount rate 0. The time dependence in (3.2) comes only through the discount factor.
The objective is to
Full rank or constraint qualifications condition holds for all arguments x(t), u(t), t, t [0,T], and
hold for all possible values of x(T) and T.

Hamiltonian function H: En x Em x En x E1 E1 is

Special Case: In the case of terminal constraint (3.6), the terminal conditions on the state and the adjoint variables in (3.11) will be, respectively,
so that the optimal control has the form
To get the adjoint equation and the multipliers associated with constraints (3.16), we form the Lagrangian: From this we get the adjoint equation
where g: En x Em xE1 Eq is continuously differentiable and terminal inequality and equality constraints:
where a: En x E1 Ela and b: En x E1 continuously differentiable.

Since x(0)=1, the control u*= x gives x= et as the solution of (3.15). Because x=et >0, it follows that u*= x > 0; thus 1 =0 from (3.20).

From (3.19) we then have 2 =1+. Substituting this into (3.18) and solving gives