Abstract
Let A be a closed subset of the tx-space E1 × En, t∈E1, x = (x1,…xn)∈En and for each (t, x)∈A, let U(t,x) be a closed subset of the u-space Em, u = (u1,…,um). We do not exclude that A coincides with the whole tx-space and that U coincides with the whole u-space. Let M denote the set of all (t, x, u) with (t, x)∈A. u∈U(t, x). Let f(t, x, u) = (f0, f) = (f0, f1,…, fn) be a continuous vector function from M into En+1. Let Bbe a closed subset of points (t1x1,t2,x2)of E2n+2, x1 = (x1 1,…x1 n, x2 = (x2 1,…x2 n.We shall consider the class of all pairs x(t), u(t), t1 ≤t≤t2, of vector functions x(t), u(t) satisfying the following conditions :
-
(a)
x(t) is absolutely continuous (AC) in [t1, t2];
-
(b)
u(t) is measurable in [t1, t2];
-
(c)
(t,x(t))∈A for every t∈[t1, t2];
-
(d)
u(t)∈U(t, x(t)) almost everywhere (a.e.) in [t1, t2];
-
(e)
f0 (t,x(t), u(t)) is L-integrable in[t1, t2];
-
(f)
dx/dt / f(t, x(t), u(t)) a.ein [t1, t2];
-
(g)
(t 1,x(t 1), t 2, x(t 2))∈B.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. Cesari, (a) Semicontinuità del calcolo delle variazioni, Annali Scuola Normale Sup. Pisa, 14, 1964, 389–423
L. Cesari, Un teorema di esistenza in problemi di controlli ottimi, Ibid, 1965, 35–78
L. Cesari, An existence theorem in problems of optimal control, J.SIAM Control, (A) 3 1965, 7–22
L. Cesari, Existence theorems for optimal solutions in Pontryagin and Lagrange problems, J. SIAM Control, (A) 3 1965, 475–498
L. Cesari, Existence theorems for generalized and usual optimal solutions in Lagrange problems with unilateral constraints. I and II. Trans. Amer Math. Soc., to appear
L. Cesari, Existence theorems for multidimensional problems of optimal control. Conference in Differential Equations, Mayaguez, Puerto Rico, December 1965, to appear.
A. F. Filippov, On certain questions in the theory of optimal control, Vestnik Moskov. Univ. Ser. Mat. Mech. Astr. 2, 1959, 25–32 (Russian). English translation in J. SIAM control (A), 1, 1962, 26–34.
R. V. Gamkrelidze, On sliding optimal regimes, Dokl. Akad. Nauk USSR 143, 1962, 1243–1245 (Russian). English translation in Soviet Math. Doklady 3, 1962, 390–395.
J. R. LaPalm, Existence analysis in problems of optimal control with closed control space and exceptional sets. A. Ph.D. thesis at the University of Michigan, Ann Arbor, Michigan, 1966.
L. Markus and E.B. Lee, Optimal control for nonlinear processes, Arch, Rational Mech. Anal. 8, 1961, 36–58.
E. J. McShane, (a) Curve-space topologies associated with variational problems, Ann. Scuola Norm. Sup. Pisa, (2) 9, 1940, 45–60. (b) Generalized curves, Duke Math. J., 6, 1940, 513–536. (c) Necessary conditions in the generalized-curve problem of the calculues of variations, Ibid., 7, 1940, 1–27 (d) Existence theorems for Bolza problems in the calculus of variations, Ibid, 28–61. (e) A metric in the space of generalized curves, Ann. of Math?, (2) 52, 1950, 328–349.
E. J. McShane, Generalized curves, Duke Math. J., 6, 1940, 513–536.
E. J. McShane, Necessary conditions in the generalized-curve problem of the calculues of variations, Ibid., 7, 1940, 1–27
E. J. McShane, Existence theorems for Bolza problems in the calculus of variations, Ibid., 1940, 28–61.
E. J. McShane, A metric in the space of generalized curves, Ann. of Math?, (2) 52, 1950, 328–349.
C.B. Morrey, (a) Multiple integral problems in the calculus of variations, Univ. of California Publ. in Math. 1, 1943, 1–30;
C.B. Morrey, Multiple integrals in the calculus of variations, Grundl, Math. Wiss. Bd. 130, Springer 1966.
M.Nagumo, Uber die gleichmässige Summierbarkeit und ihre Anwendung auf ein Variation problem. Japanese Journ, Math. 6, 1929, 173–182.
A.Plis, Trajectories and quasi trajectories of an orientor field. Bull. Acad. Pol. Sci., 11, 1963, 369–370.
L.S. Pontryagin, Optimal control processes, Uspehi Mat. Nauk 14, 1 (85), 1959, 3–20. English Translation in Amer. Math. Soc. Transl. (2), 1961, 321–339.
L.S. Pontryagin, V. C. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The mathematical theory of optimal processes, Gosudarst. Moscow 1961. English translations: Interscience 1962; Pergamon Press 1964.
E. H. Rothe, An existence theorem in the calculus of variations based on Sobolev's imbedding theorems. Archive Rat. Mech. Anal. 21., 1966, 151–162.
E. Roxin, The existence of optimal controls, Mich. Math. J., 9, 1962, 109–119.
S.Saks, Theory of integral, Monogr. Matem, 1937.
S. L. Sobolev, Applications of functional analysis in mathematical physics, Amer, Math. Soc. Translation, Providence, R. I., 1963.
L. Tonelli, Sugli integrali del calcolo delle variazioni in forma ordinaria, Annali Scuola Normale Sup. Pisa, (2) 3, 1934, 401–450. Opere Scelte, Cremonese, Roma 1962, 7 192–254;
l. Tonelli, Un teorema di calcolo delle variazioni, Rend. Accad. Lincei, 15 1932, 417–423. Opere Scelte, 3 84–91;
L. Tonelli, Fondamenti di calcolo delle variazioni, Zanichelli, Bologna 1921–23.
L. Turner, The direct method in the calculus of variations, A. Ph.D. thesis at Purdue University, Lafayette, Indiana, 1957.
A, Turowicz, Sur les trajectoires et quasi trajectoires des systèmes de commande non linéaires. Bull. Acad. Pol. Sci. 10, 1962, 529–531
A, Turowicz, Sur les zones d'émission des trajectoirės et des quasi trajectoires des systèmes de commande non linéaires. Ibid., 11, 1963, 47–50.
J. Warga, Relaxed, variational problems, J. Math. Anal. Appl.4 1962, 111–128.
T. Wazewski, Sur une généralization de la notion des solutions d'une équation au contingent, Bull. Acad. Polon. Sci.10, 1962 11–15
T. Wazewski, Sur les systèmes de commande non linéaires dont le contredomaine de command n'est pas forcément convexe, Ibid., 10, 1962, 17–21.
L. C. Young, Generalizedcurves and the calculus of variations, C.R.Soc. Sci. Varsovie, (3),30, 1937, 212–234.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Cesari, L. (2010). Existence Theorems for Lagrange and Pontryagin Problems of The Calculus of Variations and Optimal Control. More Dimensional Extensions in Sobolev Spaces. In: Conti, R. (eds) Calculus of Variations, Classical and Modern. C.I.M.E. Summer Schools, vol 39. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11042-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-11042-9_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11041-2
Online ISBN: 978-3-642-11042-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)