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A notion of $L^p$-exact controllability is introduced for linear controlled (forward) stochastic differential equations with random coefficients. Several sufficient conditions are established for such kind of exact controllability. Further, it is proved that the $L^p$-exact controllability, the validity of an observability inequality for the adjoint equation, the solvability of an optimization problem, and the solvability of an $L^p$-type norm optimal control problem are all equivalent.
Prof. Russell's lifetime love and interests are control theory and partial differential equations. We have asked Goong Chen, a former PhD student of Prof. Russell, to write a survey of his career and mathematical work. Moreover, as a way to pay our tribute to him, it is only proper here that we put together a collection of papers by researchers who work on these and closely related subjects of Prof. Russell's. Indeed, the topics collected are extensive, including the theory and applications of analysis, computation, control, optimization, etc., for various systems arising from conservation laws and quasilinear hyperbolic systems, fluids, mathematical finance, neural networks, reaction-diffusion, structural dynamics, stochastic PDEs, thermoelasticity, viscoelasticity, etc. They are covered in great depth by the authors. There are a total of 35 papers finally accepted for publication after going through strict manuscript reviews and revision. Due to the large number of papers on hand, we divide them into Groups I (20 papers) and II (15 papers). Papers in Group I are published in Discrete and Continuous Dynamical Systems-Series B (DCDS-B) Volume 14, No. 4 (2010), while those in Group II are published in Journal of Systems Science and Complexity (JSSC) Volume 23, No. 3 (2010). This division is up to the preference of the authors. We thank all the authors and reviewers for their critical contributions and valuable assistance that make the publication of these two journal issues possible.
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An optimal control problem for a semilinear elliptic equation of divergenceform is considered. Both the leading term and the semilinear term of the state equationcontain the control. The well-known Pontryagin type maximum principle for the optimal controls is the first-order necessary condition. When such a first-order necessary condition is singular in some sense, certain type of the second-order necessary condition will come in naturally. The aim of this paper is to explore such kind of conditions for our optimal control problem.
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