Solution differentiability for parametric nonlinear control problems with

SOLUTION DIFFERENTIABILITY

FOR PARAMETRIC

NONLINEAR

CONTROL PROBLEMS WITH INEQUALITY CONSTRAINTS

Helmut Maurer and Hans Josef Pesch Westf~lische Wilhelms-Universit£t M/inster, Institut fiir Numerische und instrumentelle Mathematik, Einsteinstrasse 62, 48149 Miinster, Germany. Technische Universit£t Miinchen, Mathematisches Institut, Arcisstrasse 21, 80333 Miinchen, Germany.

A b s t r a c t : This paper considers parametric nonlinear control problems subject to mixed control-state constraints. The data perturbations are modeled by a parameter p of a Banaeh space. Using recent second-order sufficient conditions (SSC) it is shown that the optimal solution and the adjoint multipliers are differentiable functions of the parameter. The proof blends numerical shooting techniques for solving the associated boundary value problem with theoretical methods for obtaining SSC. In a first step, a differentiable family of extremals for the underlying parametric boundary value problem is constructed by assuming the regularity of the shooting matrix. Optimality of this family of extremals can be established in a second step when SSC are imposed. Key Words: Parametric control problems, mixed control-state constraints, secondorder sufficient conditions, multipoint boundary value problems, shooting techniques, Riccati equation.

1

Introduction

This paper is concerned with parametric nonlinear control problems subject to mixed control-state constraints. The following parametric control problem will be referred to as OC(p) where p is a parameter belonging to a Banach space P: minimize the functional b

z(x, u,p) = f L(x(t), u(t), p)dt

(1)

subject to ~:(t)

=

f(x(t),u(t),p)

x(a)

=

V(P),

c(x(t),u(t),p)

for a.e.

x(b) = ~(p) < o

a.e.

t ~ [a,b] , ,

t e [~,bl.

(2) (3) (4)

438

We shall not treat the most general case and assume that the control variable u and the inequality constraint (4) are scalar. Extensions to the vector-valued case are possible. The functions L : / R ~+1 x P --~/R, f : / R *+1 × P - + / R ~, ~, ¢ : P - + / R " and C : / R ~+1 × P --~/R are assumed to be C2-functions on appropriate open sets. The admissible class is that of piecewise continuous control functions. Later on conditions will be imposed such that the optimal control is continuous and piecewise of class C 1. The problem OC(po) corresponding to a fixed parameter P0 • P is considered as the unperturbed or nominal problem. It will be ensured by second-order sufficient conditions (SSC) that OC(po) has a local minimum x0(t), u0(t), A0(t) where ~0(t) denotes the adjoint function which will be defined below. Our aim is to embed the unperturbed solution into a piecewise Cl-family of optimal solutions x(t,p), u(t,p), )~(t,p) for the perturbed problem OC(p) with p in a neighborhood of P0. This solution differentiability problem has been considered for pure control constraints in [2] - [5] using optimization techniques in Hilbert spaces or Bana~h spaces. The approach developed in this paper is different and melts finite-dimensional numerical solution techniques with recent theoretical results for weak second-order sufficient conditions. Solution differentiability is obtained in two steps. In a first step, a C1-family of extremals x(t, p), £(t, p) is constructed which satisfies the first order necessary conditions for OCt). This is achieved by setting up a suitable parametric boundary value problem and by imposing the regularity of the Jacobia~ for the shooting method. The second step consists in showing that this Cl-family of extremals is indeed optimal by requiring second-order sufficient conditions. Both steps are connected by the fact that a direct line can be traced from the variational system of the unperturbed boundary value problem to SSC by introducing a Riccati ODE. Some aspects of the first step have been considered already in [1], [10], [11]. Only the main results are given in this paper. Detailed proofs and more advanced numerical examples will appear elsewhere.

2

The

parametric

boundary

value

problem

f o r OC(p)

Necessary optimality conditions for Control problems with mixed constraints can be found in [8]. The Hamiltonian for the unconstrained problem (1) - (3) is

H(x,~,u,p) = L(=,~,p) + ~*/(~, u, p),

~ e ~",

(5)

whereas the augmented Hamiltonian for the constrained problem OC(p) is defined

by [Z(~,~,~,~,p) = H(=,~,~,p) + ~ C ( x , ~ , p ) ,

~ • ~.

(6)

439

The adjoint function A : [a, b] - + / R ~ and the multiplier function # : [a, b] --+/R with # > 0 and # C = 0 are determined by a suitable boundary value problem (BVP) which we shall set up now. First, a careful list of assumptions is given which is needed for a suitable numerical analysis of the problem in conjunction with SSC. The structure of the unperturbed

s o l u t i o n (x0, u0):

The active set or boundary of the inequality constraint C(x, u, po) c>O (b)

for

a_ 0 for t o < t < t o where the multiplier #o is defined by (10), (c) the Riccati ODE (23) has a finite symmetric solution Q(t) in [a, b]. Then (x0, u0) provides a local minimum/or OC(po). Moreover, uo(t) is continuous and is a el-function for t ~ t o (i = 1, 2) while xo and ~o are Cl-functions in [a, b].

Now we can state the main result of this paper.

Theorem 2: (Solution differentiability) Let (xo, uo) be feasible for OC(po) with the boundary structure (7). Let (xo,)~o) be a solution of B V P ( p o ) such that the following assumptions hold: (a) (A1) - (A3) are satisfied, (b) the multiplier #o in (10) satisfies the strict complementarity condition #o(t) > 0 for t o < t < t °, (c) the Riccati ODE (23) has a finite symmetric solution Q(t) in [a, b]. (d) the n × ~ mat~x y(b) is regular when y(t) and ~(t) are solutions of (20) with initial conditions y(a) = On, ~?(a) = In. Then there exist a neighborhood V C P of p = Po and Cl-functions

x,~,:[,~,b]xV~", t , : V - ~

(i=1,2)

and a function u : [a,b] × V -+ lR which is of class C 1 for t ¢ t~(p) (i = 1, 2), such that the following statements hold: (1) x(t,po) = xo(t), )~(t,po) = ~0(t) and u(t,po) = uo(t) for t e [a,b], (2) the gacobian of F in (19) at (so,po) is regular and x(t,p) and )~(t,p) solve BVP(p),

444

(3) the triple x( . ,p), A(. ,p), u( . ,p) satisfies the second-order sufficient conditions in Theorem 1 for every p e V and hence the pair x( . ,p), u( . ,p) provides a local minimum for OC(p). The solution differentiability provides a theoretical basis for performing a sensitivity analysis where the perturbed solution is approximated by a first order Taylor expansion according to

Ox x(t,p) ~. xo(t) + ~ ( t , p o ) ( p - p o ) , The "variations"

OA x(t,p) ~ ~o(t) + ~ ( t , p o ) ( p - p o ) .

Ox y(t) := N ( t , p 0 ) ,

OA ~(t):= N ( t , p 0 )

(26)

axe n x n-matrices of class C 1. By differentiating the boundary value problem (13) (16) one obtains a linear inhomogeneous B V P for y(t) and ~?(t).

4

An

example

with

a perturbed

control

constraint

Consider the following non-convex control problem with perturbation p E / R : minimize 1

f (u 2 - lOx2ldt

(27)

0

subject to

= x 2- u,

x(0) = 1,

x(1) = 1,

x + u < p.

(28) (29)

By studying the unconstrained problem (27), (28) first, it can be seen that the constraint (29) becomes active for the nominal parameter p = P0 -- 5.9. The unperturbed 0 0 solution (x0, Uo) has one boundary arc It1, t2] with 0 < t o < t o < 1. The BVP (13)(16) is evaluated as

(~, ~) =

{

(x 2 - 0.5x, 2x(10 - x)) , t ¢ [tl,t2] / (x 2 + x - po, 2x(10 - ~) - ~) , t e [tl,t~]

tt = A + 2(x - P0)

x(0) = 1, x(1) = 1 ,

x(t,) +0.5~(t,) = Po

(i --- 1, 2 ) .

The shooting procedure yields Ao(O) = -5.324898490, t o = 0.6735245190, t o = 0.8988553586.

(30)

445

Assumptions (A1) and (A2) are trivially satisfied whereas assumption (A3) holds with C(x, u,p) = x + u - p and C0(t °) = 1.848039743,

C(t °) = -1.673196320.

It can easily be verified that the Riccati equation (23) =

-4=o(t)q + 2 0 - 2~0(t) + 0.bQ ~ , t ¢ [tl,t~] 0 0 ( - 2 - 4 = 0 ( t ) ) q + 18 - 2~0(t) , t e [t° , t °]

has a bounded solution in [0,1]. Hence all assumptions for Theorem 2 are met and we can embed (xo, uo) into a CLfamily x(t,p),u(t,p) of solutions to the perturbed problem (27)- (29). The "variations" y(t) = Ox(t,po)/Op and 71(t) = OA(t,po)/Op in (26) satisfy a linear inhomogene0us B V P which is obtained by differentiating (30). The solution is y(0) = 0, ~/(0) = -0.2717521464 which can be used to compute dt z , ~ -~ptPO) = 0.4330244699,

dr2 -~p (Po) = -0.4869079015.

References

[1] BOCK, H.G., Zur numerischen Behandlung zustandsbeschrbnkter Steuerungsprobleme mit Mehrzielmethode und Homotopierver]ahren, ZAMM, Vol. 57, pp. T 266 - T 268, 1977. [2] DONTCHEV, A.L., HAGER, W.W., POOR.E, A.B., and YANG, B., Optimality, Stability and Convergence in Nonlinear Control, preprint, June 1992. [3] ITO, K., and KUNISCH, K., Sensitivity Analysis of Solutions to Optimization Problems in Hilbert Spaces with Applications to Optimal Control and Estimation, Journal of Differential Equations, Vol. 99, pp. 1-40, 1992. [4] MALANOWSKI, K., Second-Order Conditions and Constraint Qualifications in Stability and Sensitivity Analysis of Solutions to Optimization Problems in Hilbert Spaces, Applied Mathematics and Optimization, Vol. 25, pp. 51-79, 1992. [5] MALANOWSKI, K., Regularity of Solutions in Stability Analysis of Optimization and Optimal Control Problems, Working Papers No. 1/93, Systems Research Institute, Polish Academy of Sciences, 1993. [6] MAURER, H., First and Second Order Sufficient Optimality Conditions in Mathematical Programming and Optimal Contro~ Mathematical Programming Study, Vol. 14, pp. 163-177, 1981.

446

[7] MAURER, H., The Two-Norm Approach for Second-Order Sufficient Conditions in Mathematical Programming and Optimal Control, Preprints "Angewandte Mathematik und Informatik" der Universitiit Miinster, Report No. 6/92-N, 1992. [8] NEUSTADT, L.W., Optimization: A Theory of Necessary Conditions, Princeton University Press, Princeton, 1976. [9] ORRELL, D., and ZEIDAN, V., Another Jacobi Sufficiency Criterion for Optimal Control with Smooth Constraints, Journal of Optimization Theory and Applications, Vol. 58, pp. 283-300, 1988. [10] PESCH, H.J., Real-Time Computation of Feedback Controls for Constrained Optimal Control Problems, Part i: Neighbouring Extremals, Optimal Control Applications & Methods, Vol. 10, pp. 129-145, 1989. [11] PESCH, H.J., Real-Time Computation of Feedback Controls for Constrained Optimal Control Problems, Part 2: A Correction Method Based on Multiple Shooting, Optimal Control Applications & Methods, Vol. 10, pp. 147-171, 1989. [12] PICKENHAIN, S., Sufficiency Conditions for Weak Local Minima in Multidimensional Optimal Control Problems with Mixed Control - State Restrictions, Zeitschrift fiir Analysis und ihre Anwendungen, 1992. [13] ZEIDAN, V., Sufficiency Criteria via Focal Points and via Coupled Points, SIAM 3. Control and Optimization, Vol. 30, pp. 82-98, 1992.