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Sufficient conditions in free final time optimal control problems. A comment
JOURNAL
OF ECONOMIC
32, 367-370 (1984)
THEORY
Sufficient Conditions in Free Final Time Optimal Control Problems. A Comment ATLE SEIERSTAD* Institute
of Economics,
University
of Oslo, Boks
1095,
Blindern,
Oslo 3. Norway
Received July 27, 1982; revised January 18, 1983
By an application of sufficient conditions, assume that an optima! pair (xi(.), u,(.)) and an adjoint function p,(.) were found in the control problem in question with the final time t fixed but arbitrary. Then a sufftcient condition for one of these pairs, say x~,(.), r+(.) to be optimal in the corresponding free final time problem is that the Hamiltonian, with (x,(t), a,(,-), p,(s), 7.) inserted, is nonnegative (nonpositive) to the left (right) of r*. Journal ofEconomic Literature Classification Number: 213.
This note comments upon a sufficient condition for free time problems proposed in this journal (see [3]), which turns out not to be correct, and then proposes another sufficient condition. Consider the control problem maximize
fI f’@(t),
I’
u(t), t) dt,
r E I=
IT19Q],
where to and r1 and z2 are fixed points, to < 7, < z2. ,-?=f(x,
u(t), t),
u(t) E u,
x(t,) = x0, x0 fixed in R ‘,
U a fixed bounded subset of R’,
Xi(T) = xf )
i = l,..., 1,
xi(z) > Xi,)
i = 1 + l,..., m,
x’(z)
i = m + I,..., n.
free,
* I have benefited from comments kindly made by the referees.
367 0522-053 t/84 $3.05 All
Copyright 0 1984 by Academic Press, Inc. rights of reproduction in any form reserved.
368
ATLE
SEIERSTAD
The optimization problem consists in choosing a point r E 1, and a piecewise continuous function u(a) and a continuous and piecewise continuously differentiable function x(.), both defined on [to, r], in such a way that the triple (x(o), u(.), z) is admissible (i.e., satisfies (2), (3) and (4) and such that it maximizes the integral in (1). The functions f”, f, f I, f, f ,” and ft are assumed to be continuous on R" x R' x R (f” takes values in R, f in R", f z, f,, f y,f, are partial derivatives with respect to x and t, assumed to exist for all (x, 24,t)). If (.Z?(*), C(s), r”) is an optimal triple in the above problem, it satisfies the following necessary conditions: For some number p. > 0, and some function p(t), (po,p(t)) # 0 everywhere, we have fw),
@@>,P (4, t) = y’,”
f@(t),
u, PW, 0
for all t,
(5)
where H(x, u,p, t) =po f ‘( x, U, t) +pf (x, u, t). The function p(.) satisfies
rj = --H,(-f(t), w>, PW, 4 at all points t of continuity
of ti(.). Finally, no condition,
Pi@“) pi(z*) > 0
(=0 if Xi(r*)
> xf),
pi@“) = 0, H(lqz”),
(6)
zqz*-),p(t*),
i = l,..., I, i = I + l,..., m,
(7)
i = m + l,..., IZ, z*)[z - z*] < 0
for all
z E I.
(8)
When two additional conditions are added to (5~(8), we get a set of conditions that are sufficient in the problem where r is kept fixed equal to r* (see [5]):
PO= 1, H*(x,p(t),
(9)
t) = sup H(X, u,p(t), t) is a concave function of x, for each t. ueu
(10) In the situation where I = (to, co) (and I = m), it is asserted in [3] that if the condition H*(.%(t),p(t), t) > 0 for t E [to, r*), H*(T(z*),p(z*), t) < 0 for t E (TX, co) is added to conditions (5~(lo), then a set of sufficient conditions are obtained. The following example shows this to be incorrect, and the error in Robson’s proof is essentially that this function M(x, iji, t) fails to have the asserted concavity property. (In [3] both the equalities at the bottom of p. 441 are sometimes wrong.)
SUFFICIENT CONDITIONS
369
EXAMPLE.
max T (t - 1) x dt s.t. xY= 1, x(0) = -1, x(r) free, r E (0, 00). i’0 We readily verify that 2(t) = t - 1, r* = 1, satisfy the suffkient conditions proposed in [3]. (p(t) = - f(t - l)‘, M*(Z(t),p(t), t) = i(t - I)“, while H*(Z(r*),p(z*), t) = (t - 1) X(7*) = 0). However, this solution is not optimal. (E.g., r = 2 gives the criterion a higher value.) Below another sufficient condition is proposed: THEOREM. Assume that for each 5 E I p,(.)) defined on [t,, T] satisfying (2)--(7), r Jixed). Assume moreover, that when x,(.) 120other function than p,(a) satisfying (5),
there exists a triple (x,4.), u,(*), (91, (10) (i.e., q(.) is optimal for and u,(.) are given, there exists (6), (9) on [to, 21.
Assume aho that the function z+ X,(T) is continuous and piece&e continuously difSerentiable. Assumefinally that the jktion
has the property that there exists a T* E I, such that d(r)>0
for T < z*,
if
7, < 7+,
d(r)<0
for 5 > T*,
if
f2 > 7*.
(12)
Then the triple (x,*(.), us*(.), T*) is optimal. Remark.
The Theorem is also valid if I = [rI, co).
Proof. Let W(y, r) be the supremum of the value of the criterion for all pairs (x(a), u(.)) satisfying (2), with x(t) ending at y for t = 7, and let V(r) = w?,(7), 712 7 E [ rl, ~~1.By the uniquenessand concavity assumptionsof the Theorem, W( y, 7) is defined in a neighbourhood of (x,(t), z), and is differentiable at this point, with derivative (-p,(z), d(r)), see (4, Remark]. Thus V(a) is continuous and piecewise differentiable, with + 47)
(13)
existing at all r E (z,, r2) at which x,(z) is differentiable. By (7), note that -X,(T)] > 0 for any r’ E I. Dividing by r’ - r < 0 (respectively, z’ - T > O), we get p,(z)(dx,(z)/dT) < 0 (resp. >O). Thus p,(z)(dx,(z)/dz) = 0. For points 7 of the above type, we then get dV(s)/dz = d(z). Finally, using the two inequalities in (12), we get V(z) < V(r*) both for z < r* and r > r*.
~,(z)[x,,(z’)
370
ATLE SEIERSTAD
Note 1. In [6] the above Theorem is generalized to problems with other types of side conditions and it is also shown that the uniqueness condition can be dropped. Other types of sufficient conditions for free final time have been given in [l] (see also [2]). Note 2. In the Example the function evidently does not satisfy (12).
d(z) becomes i(z - I)“, which
REFERENCES 1. P. M. MEREAU AND W. F. POWERS,A direct sufficient condition for free final time optimal control problems, SZIAM J. Control 14 (1976), 613-622. 2. D. W. PETERSON AND J. H. ZALKIND, A review of direct sufficient conditions in optimal control theory, Inter-nut. J. Control 28, No. 4 (1981), 589-610. 9 A. J. ROBSON, Sufficiency of the Pontryagin conditions for optimal control when the time horizon is free, J. Econ. Theory 24 (1981), 438-445. 4. A. SEIERSTAD, Differentiability properties of the optimal value function in control theory, J. Econ. Dynamic Control 4 (1982), 303-310. 5. A. SEIERSTAD AND K. SYDSBTER, Sufficient conditions in optimal control theory, Infernat. Econ. Review 18, No. 2 (1977), 367-391. 6. A. SEIERSTAD, Sufficient conditions in free final time optimal control problems, Memorandum from Institute of Economics, University of Oslo, January 15, 1984.
OF ECONOMIC
32, 367-370 (1984)
THEORY
Sufficient Conditions in Free Final Time Optimal Control Problems. A Comment ATLE SEIERSTAD* Institute
of Economics,
University
of Oslo, Boks
1095,
Blindern,
Oslo 3. Norway
Received July 27, 1982; revised January 18, 1983
By an application of sufficient conditions, assume that an optima! pair (xi(.), u,(.)) and an adjoint function p,(.) were found in the control problem in question with the final time t fixed but arbitrary. Then a sufftcient condition for one of these pairs, say x~,(.), r+(.) to be optimal in the corresponding free final time problem is that the Hamiltonian, with (x,(t), a,(,-), p,(s), 7.) inserted, is nonnegative (nonpositive) to the left (right) of r*. Journal ofEconomic Literature Classification Number: 213.
This note comments upon a sufficient condition for free time problems proposed in this journal (see [3]), which turns out not to be correct, and then proposes another sufficient condition. Consider the control problem maximize
fI f’@(t),
I’
u(t), t) dt,
r E I=
IT19Q],
where to and r1 and z2 are fixed points, to < 7, < z2. ,-?=f(x,
u(t), t),
u(t) E u,
x(t,) = x0, x0 fixed in R ‘,
U a fixed bounded subset of R’,
Xi(T) = xf )
i = l,..., 1,
xi(z) > Xi,)
i = 1 + l,..., m,
x’(z)
i = m + I,..., n.
free,
* I have benefited from comments kindly made by the referees.
367 0522-053 t/84 $3.05 All
Copyright 0 1984 by Academic Press, Inc. rights of reproduction in any form reserved.
368
ATLE
SEIERSTAD
The optimization problem consists in choosing a point r E 1, and a piecewise continuous function u(a) and a continuous and piecewise continuously differentiable function x(.), both defined on [to, r], in such a way that the triple (x(o), u(.), z) is admissible (i.e., satisfies (2), (3) and (4) and such that it maximizes the integral in (1). The functions f”, f, f I, f, f ,” and ft are assumed to be continuous on R" x R' x R (f” takes values in R, f in R", f z, f,, f y,f, are partial derivatives with respect to x and t, assumed to exist for all (x, 24,t)). If (.Z?(*), C(s), r”) is an optimal triple in the above problem, it satisfies the following necessary conditions: For some number p. > 0, and some function p(t), (po,p(t)) # 0 everywhere, we have fw),
@@>,P (4, t) = y’,”
f@(t),
u, PW, 0
for all t,
(5)
where H(x, u,p, t) =po f ‘( x, U, t) +pf (x, u, t). The function p(.) satisfies
rj = --H,(-f(t), w>, PW, 4 at all points t of continuity
of ti(.). Finally, no condition,
Pi@“) pi(z*) > 0
(=0 if Xi(r*)
> xf),
pi@“) = 0, H(lqz”),
(6)
zqz*-),p(t*),
i = l,..., I, i = I + l,..., m,
(7)
i = m + l,..., IZ, z*)[z - z*] < 0
for all
z E I.
(8)
When two additional conditions are added to (5~(8), we get a set of conditions that are sufficient in the problem where r is kept fixed equal to r* (see [5]):
PO= 1, H*(x,p(t),
(9)
t) = sup H(X, u,p(t), t) is a concave function of x, for each t. ueu
(10) In the situation where I = (to, co) (and I = m), it is asserted in [3] that if the condition H*(.%(t),p(t), t) > 0 for t E [to, r*), H*(T(z*),p(z*), t) < 0 for t E (TX, co) is added to conditions (5~(lo), then a set of sufficient conditions are obtained. The following example shows this to be incorrect, and the error in Robson’s proof is essentially that this function M(x, iji, t) fails to have the asserted concavity property. (In [3] both the equalities at the bottom of p. 441 are sometimes wrong.)
SUFFICIENT CONDITIONS
369
EXAMPLE.
max T (t - 1) x dt s.t. xY= 1, x(0) = -1, x(r) free, r E (0, 00). i’0 We readily verify that 2(t) = t - 1, r* = 1, satisfy the suffkient conditions proposed in [3]. (p(t) = - f(t - l)‘, M*(Z(t),p(t), t) = i(t - I)“, while H*(Z(r*),p(z*), t) = (t - 1) X(7*) = 0). However, this solution is not optimal. (E.g., r = 2 gives the criterion a higher value.) Below another sufficient condition is proposed: THEOREM. Assume that for each 5 E I p,(.)) defined on [t,, T] satisfying (2)--(7), r Jixed). Assume moreover, that when x,(.) 120other function than p,(a) satisfying (5),
there exists a triple (x,4.), u,(*), (91, (10) (i.e., q(.) is optimal for and u,(.) are given, there exists (6), (9) on [to, 21.
Assume aho that the function z+ X,(T) is continuous and piece&e continuously difSerentiable. Assumefinally that the jktion
has the property that there exists a T* E I, such that d(r)>0
for T < z*,
if
7, < 7+,
d(r)<0
for 5 > T*,
if
f2 > 7*.
(12)
Then the triple (x,*(.), us*(.), T*) is optimal. Remark.
The Theorem is also valid if I = [rI, co).
Proof. Let W(y, r) be the supremum of the value of the criterion for all pairs (x(a), u(.)) satisfying (2), with x(t) ending at y for t = 7, and let V(r) = w?,(7), 712 7 E [ rl, ~~1.By the uniquenessand concavity assumptionsof the Theorem, W( y, 7) is defined in a neighbourhood of (x,(t), z), and is differentiable at this point, with derivative (-p,(z), d(r)), see (4, Remark]. Thus V(a) is continuous and piecewise differentiable, with + 47)
(13)
existing at all r E (z,, r2) at which x,(z) is differentiable. By (7), note that -X,(T)] > 0 for any r’ E I. Dividing by r’ - r < 0 (respectively, z’ - T > O), we get p,(z)(dx,(z)/dT) < 0 (resp. >O). Thus p,(z)(dx,(z)/dz) = 0. For points 7 of the above type, we then get dV(s)/dz = d(z). Finally, using the two inequalities in (12), we get V(z) < V(r*) both for z < r* and r > r*.
~,(z)[x,,(z’)
370
ATLE SEIERSTAD
Note 1. In [6] the above Theorem is generalized to problems with other types of side conditions and it is also shown that the uniqueness condition can be dropped. Other types of sufficient conditions for free final time have been given in [l] (see also [2]). Note 2. In the Example the function evidently does not satisfy (12).
d(z) becomes i(z - I)“, which
REFERENCES 1. P. M. MEREAU AND W. F. POWERS,A direct sufficient condition for free final time optimal control problems, SZIAM J. Control 14 (1976), 613-622. 2. D. W. PETERSON AND J. H. ZALKIND, A review of direct sufficient conditions in optimal control theory, Inter-nut. J. Control 28, No. 4 (1981), 589-610. 9 A. J. ROBSON, Sufficiency of the Pontryagin conditions for optimal control when the time horizon is free, J. Econ. Theory 24 (1981), 438-445. 4. A. SEIERSTAD, Differentiability properties of the optimal value function in control theory, J. Econ. Dynamic Control 4 (1982), 303-310. 5. A. SEIERSTAD AND K. SYDSBTER, Sufficient conditions in optimal control theory, Infernat. Econ. Review 18, No. 2 (1977), 367-391. 6. A. SEIERSTAD, Sufficient conditions in free final time optimal control problems, Memorandum from Institute of Economics, University of Oslo, January 15, 1984.