zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Optimal control of Sobolev type linear equations. (English. Russian original) Zbl 1095.49004
The paper concerns minimization of the cost functional $$J(u)=\Vert Cx(\tau)-z_0(\tau)\Vert^2+\sum_0^{p+1}\int_0^\tau \langle N_qu^{(q)}(t),u^{(q)}(t)\rangle\,dt$$ subject to the Cauchy system $L\dot{x}(t)=Mx(t)+y(t)+Bu(t)$, $x(0)=x_0$. Here the linear, continuous operator $L$ and the linear, closed, densely defined operator $M$ generate a strongly continuous semigroup for the homogeneous equation $L\dot{x}(t)=Mx(t)$ as well as a certain nilpotent operator of degree $p$, at most [see {\it V. E. Fedorov}, “Degenerate strongly continuous semigroup of operators”, St. Petersbg. Math. J. 12, No. 3, 471--489 (2001); translation from Algebra Anal. 12, No. 3, 173--200 (2001; Zbl 0988.47027)], whereas the control $u$ belongs to a Sobolev-Bochner space $W_2^{p+1}$. The main results establish existence and uniqueness of a strong solution $x$ to the Cauchy system, and existence of an optimal control $u$ in case of certain control sets (possibly depending on $y$ and $x_0$). The results are applied to a class of initial-boundary value problems for partial differential equations.
MSC:
49J20Optimal control problems with PDE (existence)
49K15Optimal control problems with ODE (optimality conditions)
47D06One-parameter semigroups and linear evolution equations
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
WorldCat.org
Full Text: DOI
References:
[1] Sviridyuk, G.A., Differents. Uravn., 1987, vol. 23, no. 12, pp. 2168-2171.
[2] Lions, J.-L., Contr?le optimal de syst?mes gouvern?s par des ?quations aux d?riv?es partielles, Paris: Dunod, 1968. Translated under the title Optimal?noe upravlenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi, Moscow: Mir, 1972.
[3] Balakrishnan, A.V., Applied Functional Analysis, New York: Springer, 1976. Translated under the title Prikladnoi funktsional?nyi analiz, Moscow: Nauka, 1980.
[4] Sviridyuk, G.A. and Efremov, A.A., Differents. Uravn., 1995, vol. 31, no. 11, pp. 1912-1919.
[5] Sviridyuk, G.A. and Efremov, A.A., Izv. Vyssh. Uchebn. Zaved. Matematika, 1996, no. 12, pp. 75-83.
[6] Sviridyuk, G.A. and Efremov, A.A., Dokl. RAN, 1999, vol. 364, no. 3, pp. 323-325.
[7] Fedorov, V.E., Dokl. RAN, 1996, vol. 351, no. 3, pp. 316-318.
[8] Fedorov, V.E., Algebra i Analiz, 2000, vol. 12, no. 3, pp. 173-200.
[9] Sviridyuk, G.A., Uspekhi Mat. Nauk, 1994, vol. 49, no. 4, pp. 47-74.
[10] Boyarintsev, Yu. E. and Chistyakov, V.F., Algebro-differentsial?nye sistemy. Metody resheniya i issledovaniya (Algebraic Differential Systems. Solution and Investigation Methods), Novosibirsk, 1998.
[11] Lewis, F.L., Circuits, Systems and Signal Processing, 1986, vol. 5, no. 1, pp. 3-36. · Zbl 0613.93029 · doi:10.1007/BF01600184
[12] Kurina, G.A., Izv. RAN. Tekhn. Kibernetika, 1992, no. 4, pp. 20-48.
[13] Kurina, G.A., Mat. Zametki, 2001, vol. 70, no. 2, pp. 230-236.
[14] Fedorov, V.E., Polugruppy i gruppy operatorov s yadrami (Semigroup and Groups of Operators with Kernels), Chelyabinsk, 1998.
[15] Triebel, H., Interpolation Theory, Function Spaces, Differential Operators, Berlin: Birkh?user, 1977. Translated under the title Teoriya interpolyatsii, funktsional?nye prostranstva, differentsial?nye operatory, Moscow: Mir, 1980.