When a differential field <em>K</em> having <em>n</em> commuting derivations is given together with two finitely generated differential extensions <em>L</em> and <em>M</em&...When a differential field <em>K</em> having <em>n</em> commuting derivations is given together with two finitely generated differential extensions <em>L</em> and <em>M</em> of <em>K</em>, an important problem in differential algebra is to exhibit a common differential extension <em>N</em> in order to define the new differential extensions <em>L</em><span style="white-space:nowrap;"><span style="white-space:nowrap;">∩</span></span><em>M </em>and the smallest differential field <span style="white-space:nowrap;">(<em>L</em>,<em>M</em> ) <span style="white-space:nowrap;"><span style="white-space:nowrap;">⊂</span></span> <em>N</em></span> containing both <em>L</em> and <em>M</em>. Such a result allows to generalize the use of complex numbers in classical algebra. Having now two finitely generated differential modules<em> L</em> and <em>M</em> over the non-commutative ring <span style="white-space:nowrap;"><em>D</em> = <em>K </em>[<em>d</em><sub>1</sub>,...,<em>d</em><sub>n</sub>] = <em>K</em> [<em>d</em>]</span> of differential operators with coefficients in <em>K</em>, we may similarly look for a differential module <em>N</em> containing both <em>L</em> and <em>M </em>in order to define <span style="white-space:nowrap;"><em>L</em>∩<em>M</em></span> and <span style="white-space:nowrap;"><em>L</em>+<em>M</em></span>. This is <em>exactly</em> the situation met in linear or non-linear OD or PD control theory by selecting the inputs and the outputs among the control variables. However, in many recent books and papers, we have shown that controllability was a <em>built-in</em> property of a control system, not depending on the choice of inputs and outputs. The purpose of this paper is thus to revisit control theory by showing the specific importance of the two previous problems and the part plaid by <em>N</em> in both cases for the parametrization of the control system. An important tool will be the study of <em>differential correspondence</em><em>s</em>, a modern name for what was called <em>B<span style="white-space:nowrap;"><span style="white-space:nowrap;">ä</span></span>cklund problem</em> during the last century, namely the elimination theory for groups of variables among systems of linear or nonlinear OD or PD equations. The main difficulty is to revisit <em>differential homological algebra</em> by using noncommutative localization as a way to generalize the symbolic calculus in the style of Heaviside and Mikusinski. Finally, when <em>M</em> is a <em>D</em>-module, this paper is using for the first time the fact that the system <span style="white-space:nowrap;"><em>R</em> = <em>hom<sub>K</sub></em> (<em>M</em>,<em>K</em>)</span> is a <em>D</em>-module for the Spencer operator acting on sections, avoiding thus behaviours, trajectories and signal spaces in a purely formal way, contrary to a few recent works on this difficult subject.展开更多
This paper deals with the structural analysis problem of dynamic lumped process high-index differential algebraic equations(DAE)models.The existing graph theoretical method depends on the change in the relative positi...This paper deals with the structural analysis problem of dynamic lumped process high-index differential algebraic equations(DAE)models.The existing graph theoretical method depends on the change in the relative position of underspecified and overspecified subgraphs and has an effect to the value of the differential index for complex models.In this paper,we consider two methods for index reduction of such models by differentiation:Pryce’s method and the symbolic differential elimination algorithm rifsimp.They can remedy the above drawbacks.Discussion and comparison of these methods are given via a class of fundamental process simulation examples.In particular,the efficiency of Pryce’s method is illustrated as a function of the number of tanks in process design.Moreover,a range of nontrivial problems are demonstrated by the symbolic differential elimination algorithm and fast prolongation.展开更多
文摘When a differential field <em>K</em> having <em>n</em> commuting derivations is given together with two finitely generated differential extensions <em>L</em> and <em>M</em> of <em>K</em>, an important problem in differential algebra is to exhibit a common differential extension <em>N</em> in order to define the new differential extensions <em>L</em><span style="white-space:nowrap;"><span style="white-space:nowrap;">∩</span></span><em>M </em>and the smallest differential field <span style="white-space:nowrap;">(<em>L</em>,<em>M</em> ) <span style="white-space:nowrap;"><span style="white-space:nowrap;">⊂</span></span> <em>N</em></span> containing both <em>L</em> and <em>M</em>. Such a result allows to generalize the use of complex numbers in classical algebra. Having now two finitely generated differential modules<em> L</em> and <em>M</em> over the non-commutative ring <span style="white-space:nowrap;"><em>D</em> = <em>K </em>[<em>d</em><sub>1</sub>,...,<em>d</em><sub>n</sub>] = <em>K</em> [<em>d</em>]</span> of differential operators with coefficients in <em>K</em>, we may similarly look for a differential module <em>N</em> containing both <em>L</em> and <em>M </em>in order to define <span style="white-space:nowrap;"><em>L</em>∩<em>M</em></span> and <span style="white-space:nowrap;"><em>L</em>+<em>M</em></span>. This is <em>exactly</em> the situation met in linear or non-linear OD or PD control theory by selecting the inputs and the outputs among the control variables. However, in many recent books and papers, we have shown that controllability was a <em>built-in</em> property of a control system, not depending on the choice of inputs and outputs. The purpose of this paper is thus to revisit control theory by showing the specific importance of the two previous problems and the part plaid by <em>N</em> in both cases for the parametrization of the control system. An important tool will be the study of <em>differential correspondence</em><em>s</em>, a modern name for what was called <em>B<span style="white-space:nowrap;"><span style="white-space:nowrap;">ä</span></span>cklund problem</em> during the last century, namely the elimination theory for groups of variables among systems of linear or nonlinear OD or PD equations. The main difficulty is to revisit <em>differential homological algebra</em> by using noncommutative localization as a way to generalize the symbolic calculus in the style of Heaviside and Mikusinski. Finally, when <em>M</em> is a <em>D</em>-module, this paper is using for the first time the fact that the system <span style="white-space:nowrap;"><em>R</em> = <em>hom<sub>K</sub></em> (<em>M</em>,<em>K</em>)</span> is a <em>D</em>-module for the Spencer operator acting on sections, avoiding thus behaviours, trajectories and signal spaces in a purely formal way, contrary to a few recent works on this difficult subject.
基金supported by the National Basic Research Program of China(2011CB302402)the National Natural Science Foundation of China(Grant Nos.91118001,11171053)+1 种基金the West Light Foundation of the Chinese Academy of Sciences,China Postdoctoral Science Foundation Funded Project(Grant No.2012M521692)the Chinese Academy of Sciences Visiting Professorship for Senior International Scientists(Grant No.2010T2G31)。
文摘This paper deals with the structural analysis problem of dynamic lumped process high-index differential algebraic equations(DAE)models.The existing graph theoretical method depends on the change in the relative position of underspecified and overspecified subgraphs and has an effect to the value of the differential index for complex models.In this paper,we consider two methods for index reduction of such models by differentiation:Pryce’s method and the symbolic differential elimination algorithm rifsimp.They can remedy the above drawbacks.Discussion and comparison of these methods are given via a class of fundamental process simulation examples.In particular,the efficiency of Pryce’s method is illustrated as a function of the number of tanks in process design.Moreover,a range of nontrivial problems are demonstrated by the symbolic differential elimination algorithm and fast prolongation.