'E.t moi, .. " si j'avait su comment en revenir, One service mathematics has rendered the human race. It has put common sense back je n'y serais point aile.' Jules Verne where it belongs, on the topmost shelf nex t to the dusty canister labelled 'discarded non- The series is divergent; thererore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Inhalt

1. Models.- 1. Formal prerequisites.- 2. Aftereffect in mechanics.- 3. Hereditary phenomena in physics.- 4. Models with delays in technical problems.- 5. Aftereffect in biology.- 6. Aftereffect in medicine.- 7. Aftereffect in economy and other sciences.- 2. General theory.- 1. Introduction. Method of steps.- 2. Cauchy problem for RDEs.- 3. Cauchy problem for NDEs.- 4. Differential inclusions of retarded type (RDIs).- 5. General linear equations with aftereffect.- 6. Linear autonomous equations.- 7. Hopf bifurcation.- 8. Stocnastic retarded differential equations (SRDEs).- 3. Stability of retarded differential equations.- 1. Liapunov's direct method.- 2. Linear autonomous equations.- 4. Stability of neutral type functional differential equations.- 1. Direct Liapunov's method.- 2. Stability of linear autonomous equations.- 5. Stability of stochastic functional differential equations.- 1. Statement of the problem.- 2. Liapunov's direct method.- 3. Boundedness of moments of solutions.- 6. Problems of control for deterministic FDEs.- 1. The dynamic programming method for deterministic equations. Bellman's equation.- 2. Linear quadratic problems.- 3. Optimal control of bilinear hereditary systems.- 4. Control problems with phase constraint formula.- 5. Necessary optimality conditions.- 7. Optimal control of stochastic delay systems.- 1. Dynamic programming method for controlled stochastic hereditary processes.- 2. The linear quadratic problem.- 3. Approximate optimal control for systems with small parameters.- 4. Another approach to the problem of optimal synthesis control.- 8. State estimates of stochastic systems with delay.- 1. Filtering of Gaussian processes.- 2. Filtering of solutions of Itô equations with delay.