{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T08:51:58Z","timestamp":1787388718027,"version":"build-2736575974"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We consider a system that is governed by a PDE of hyperbolic type, namely, the wave equation, and is controlled by a Dirichlet boundary control. For the penalization of terminal constraints, we compare the convergence of three penalization techniques: a differentiable penalty method, an exact penalization, and a smoothed exact penalization. The error of the solution of the differentiably penalized problem is at most of the order $1\/\\sqrt{\\gamma}$, where $\\gamma$ is the penalty parameter. The exact penalization yields the exact solution if $\\gamma$ is sufficiently large. If $\\gamma$ is sufficiently large for the smoothed exact penalization, we obtain an error bound of the order $1\/\\sqrt{\\beta}$, where $\\beta$ is the additional smoothing parameter. This method yields differentiable objective functions that remain uniformly strongly convex, since $\\gamma$ need not tend to infinity to obtain convergence. We also consider the penalization of distributed inequality state constraints that prescribe an upper bound for the $L^\\infty$-norm of the state. We prove the convergence with respect to the $L^2$-norm for a penalization of the constraint violation measured in the $L^1$-norm. We use this penalization technique to prove the existence of optimal controls for the inequality state constrained problem.<\/jats:p>","DOI":"10.1137\/080725921","type":"journal-article","created":{"date-parts":[[2009,11,20]],"date-time":"2009-11-20T22:10:21Z","timestamp":1258755021000},"page":"3026-3051","source":"Crossref","is-referenced-by-count":26,"title":["Penalty Techniques for State Constrained Optimal Control Problems with the Wave Equation"],"prefix":"10.1137","volume":"48","author":[{"given":"Martin","family":"Gugat","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,11,20]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012901392529"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/0324078"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202508002723"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.7151\/dmdico.1074"},{"key":"R5","unstructured":"P. G. Ciarlet,\n                      Mathematical Elasticity, Vol.\n                      I.\n                      Three-Dimensional Elasticity\n                      , Stud. Math. Appl. 20, North\u2013Holland, Amsterdam, 1988."},{"key":"R6","unstructured":"I. Ekeland and R. Temam,\n                      Convex Analysis and Variational Problems\n                      , Stud. Math. Appl. 1, North\u2013Holland, Amsterdam, 1976."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/040610222"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/j.matcom.2008.02.014"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/0313011"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"M. Gugat,\n                      $L^1$-optimal boundary control of a string to rest in finite time\n                      , in Recent Advances in Optimization, Lecture Notes in Econom. and Math. Systems 563, A. Seeger, ed., Springer-Verlag, Berlin, 2006, pp. 149\u2013162.","DOI":"10.1007\/3-540-28258-0_10"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/06065636x"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1002\/zamm.200700154"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1093\/imamci\/dnm014"},{"key":"R14","first-page":"231","volume":"28","author":"Gugat M.","year":"2009","journal-title":"Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0101-8205","issn-type":"print"},{"key":"R15","unstructured":"M. Gugat and M. Herty,\n                      The smoothed-penalty algorithm for state constrained optimal control problems for partial differential equations\n                      , Optim. Methods Softw., to appear."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1023\/A:1015472323967"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1051\/cocv:2007044"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012903419212"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-007-9011-6"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"W. Krabs,\n                      On Moment Theory and Controllability of One-Dimensional Vibrating Systems and Heating Processes\n                      , Lecture Notes in Control and Inform. Sci. 173, Springer-Verlag, Heidelberg, 1992.","DOI":"10.1007\/BFb0039513"},{"key":"R21","first-page":"531","volume":"28","author":"Lagnese J. E.","year":"1999","journal-title":"Control Cybernet.","ISSN":"https:\/\/id.crossref.org\/issn\/0324-8569","issn-type":"print"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012998333530"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-005-3056-1"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623403426519"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144503432862"},{"key":"R26","doi-asserted-by":"crossref","unstructured":"E. Zuazua,\n                      Controllability and observability of partial differential equations: Some results and open problems\n                      , in Handbook of Differential Equations: Evolutionary Differential Equations, Vol. 3, C. M. Dafermos and E. Feireisl, eds., Elsevier, New York, 2006, pp. 527\u2013621.","DOI":"10.1016\/S1874-5717(07)80010-7"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080725921","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:40:27Z","timestamp":1787316027000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080725921"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,11,20]]},"references-count":26,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/080725921"],"URL":"https:\/\/doi.org\/10.1137\/080725921","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,11,20]]}}}