Lecture 2. LINEAR DIFFERENTIAL SYSTEMS p.1/22
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1 LINEAR DIFFERENTIAL SYSTEMS Harry Trentelman University of Groningen, The Netherlands Minicourse ECC 2003 Cambridge, UK, September 2, 2003 LINEAR DIFFERENTIAL SYSTEMS p1/22
2 Part 1: Generalities LINEAR DIFFERENTIAL SYSTEMS p2/22
3 Linear differential systems We discuss the theory of dynamical systems that are LINEAR DIFFERENTIAL SYSTEMS p3/22
4 Linear differential systems We discuss the theory of dynamical systems that are 1 linear, LINEAR DIFFERENTIAL SYSTEMS p3/22
5 Linear differential systems We discuss the theory of dynamical systems that are 1 linear, meaning ; LINEAR DIFFERENTIAL SYSTEMS p3/22
6 Linear differential systems We discuss the theory of dynamical systems that are 1 linear, meaning ; 2 time-invariant, LINEAR DIFFERENTIAL SYSTEMS p3/22
7 Linear differential systems We discuss the theory of dynamical systems that are 1 linear, meaning ; 2 time-invariant, meaning where denotes the shift,, LINEAR DIFFERENTIAL SYSTEMS p3/22
8 Linear differential systems We discuss the theory of dynamical systems that are 1 linear, meaning ; 2 time-invariant, meaning where denotes the shift,, 3 differential, LINEAR DIFFERENTIAL SYSTEMS p3/22
9 Linear differential systems We discuss the theory of dynamical systems that are 1 linear, meaning ; 2 time-invariant, meaning where denotes the shift,, 3 differential, meaning consists of the solutions of a system of differential equations LINEAR DIFFERENTIAL SYSTEMS p3/22
10 Linear constant coefficient differential equations Variables: equations, up to times differentiated, Coefficients : 3 indices! LINEAR DIFFERENTIAL SYSTEMS p4/22
11 Linear constant coefficient differential equations Variables: equations, up to times differentiated, Coefficients : 3 indices! for the -th differential equation, LINEAR DIFFERENTIAL SYSTEMS p4/22
12 Linear constant coefficient differential equations Variables: equations, up to times differentiated, Coefficients : 3 indices! for the -th differential equation, for the variable involved, LINEAR DIFFERENTIAL SYSTEMS p4/22
13 Linear constant coefficient differential equations Variables: equations, up to times differentiated, Coefficients : 3 indices! for the -th differential equation, for the variable involved, for the order of differentiation LINEAR DIFFERENTIAL SYSTEMS p4/22
14 In vector/matrix notation: LINEAR DIFFERENTIAL SYSTEMS p5/22
15 In vector/matrix notation: Yields with LINEAR DIFFERENTIAL SYSTEMS p5/22
16 Combined with the polynomial matrix (in the indeterminate ) we obtain for this the short notation LINEAR DIFFERENTIAL SYSTEMS p6/22
17 Combined with the polynomial matrix (in the indeterminate ) we obtain for this the short notation Including latent variables with LINEAR DIFFERENTIAL SYSTEMS p6/22
18 Polynomial matrices A polynomial matrix is a polynomial with matrix coefficients: with LINEAR DIFFERENTIAL SYSTEMS p7/22
19 Polynomial matrices A polynomial matrix is a polynomial with matrix coefficients: with We may view also as a matrix of polynomials: with the s polynomials with real coefficients LINEAR DIFFERENTIAL SYSTEMS p7/22
20 Polynomial matrices A polynomial matrix is a polynomial with matrix coefficients: with We may view also as a matrix of polynomials: with the s polynomials with real coefficients Notation:,,, LINEAR DIFFERENTIAL SYSTEMS p7/22
21 What do we mean by the behavior of this system of differential equations? LINEAR DIFFERENTIAL SYSTEMS p8/22
22 What do we mean by the behavior of this system of differential equations? When shall we define? to be a solution of LINEAR DIFFERENTIAL SYSTEMS p8/22
23 What do we mean by the behavior of this system of differential equations? When shall we define? to be a solution of Possibilities: LINEAR DIFFERENTIAL SYSTEMS p8/22
24 What do we mean by the behavior of this system of differential equations? When shall we define? to be a solution of Possibilities: Strong solutions? LINEAR DIFFERENTIAL SYSTEMS p8/22
25 What do we mean by the behavior of this system of differential equations? When shall we define? to be a solution of Possibilities: Strong solutions? Weak solutions? LINEAR DIFFERENTIAL SYSTEMS p8/22
26 What do we mean by the behavior of this system of differential equations? When shall we define? to be a solution of Possibilities: Strong solutions? Weak solutions? (infinitely differentiable) solutions? LINEAR DIFFERENTIAL SYSTEMS p8/22
27 What do we mean by the behavior of this system of differential equations? When shall we define? to be a solution of Possibilities: Strong solutions? Weak solutions? (infinitely differentiable) solutions? Distributional solutions? LINEAR DIFFERENTIAL SYSTEMS p8/22
28 We will be pragmatic, and take the easy way out: solutions! LINEAR DIFFERENTIAL SYSTEMS p9/22
29 We will be pragmatic, and take the easy way out: solutions! -solution: is a -solution of if 1 is infinitely differentiable ( := ), and 2 LINEAR DIFFERENTIAL SYSTEMS p9/22
30 We will be pragmatic, and take the easy way out: solutions! -solution: is a -solution of if 1 is infinitely differentiable ( := ), and 2 Transmits main ideas, easier to handle, easy theory, sometimes (too) restrictive LINEAR DIFFERENTIAL SYSTEMS p9/22
31 Whence, defines the system with LINEAR DIFFERENTIAL SYSTEMS p10/22
32 Whence, defines the system with Proposition: This system is linear and time-invariant LINEAR DIFFERENTIAL SYSTEMS p10/22
33 Whence, defines the system with Proposition: This system is linear and time-invariant Note that is equal to the kernel of the operator We will therefore call or the behavior a kernel representation of this system, LINEAR DIFFERENTIAL SYSTEMS p10/22
34 Whence, defines the system with Proposition: This system is linear and time-invariant Note that is equal to the kernel of the operator We will therefore call or the behavior a kernel representation of this system, determines uniquely, the converse is not true! LINEAR DIFFERENTIAL SYSTEMS p10/22
35 Notation and nomenclature all such systems (with any - finite - number of variables) with variables (no ambiguity regarding ) LINEAR DIFFERENTIAL SYSTEMS p11/22
36 Notation and nomenclature all such systems (with any - finite - number of variables) with variables (no ambiguity regarding ) Elements of linear differential systems has behavior or : the system induced by LINEAR DIFFERENTIAL SYSTEMS p11/22
37 Part 2: Inputs and outputs LINEAR DIFFERENTIAL SYSTEMS p12/22
38 Linear differential system: LINEAR DIFFERENTIAL SYSTEMS p13/22
39 Linear differential system: induced by a polynomial matrix : variable Idea: there are degrees of freedom in the differential equation In other words: the requirement leaves some of the components unconstrained These components are arbitrary functions LINEAR DIFFERENTIAL SYSTEMS p13/22
40 Linear differential system: induced by a polynomial matrix : variable Idea: there are degrees of freedom in the differential equation In other words: the requirement leaves some of the components unconstrained These components are arbitrary functions inputs LINEAR DIFFERENTIAL SYSTEMS p13/22
41 Linear differential system: induced by a polynomial matrix : variable Idea: there are degrees of freedom in the differential equation In other words: the requirement leaves some of the components unconstrained These components are arbitrary functions inputs After choosing these free components, the remaining components are determined up to initial conditions LINEAR DIFFERENTIAL SYSTEMS p13/22
42 Linear differential system: induced by a polynomial matrix : variable Idea: there are degrees of freedom in the differential equation In other words: the requirement leaves some of the components unconstrained These components are arbitrary functions inputs After choosing these free components, the remaining components are determined up to initial conditions outputs LINEAR DIFFERENTIAL SYSTEMS p13/22
43 Linear differential system: induced by a polynomial matrix : variable Idea: there are degrees of freedom in the differential equation In other words: the requirement leaves some of the components unconstrained These components are arbitrary functions inputs After choosing these free components, the remaining components are determined up to initial conditions outputs LINEAR DIFFERENTIAL SYSTEMS p13/22
44 Example Position of point mass subject to a force : LINEAR DIFFERENTIAL SYSTEMS p14/22
45 Example Position of point mass subject to a force : Three (differential) equations, six variables does not put constraints on : is allowed to be any function After choosing, is determined up to and LINEAR DIFFERENTIAL SYSTEMS p14/22
46 Example Position of point mass subject to a force : Three (differential) equations, six variables does not put constraints on : is allowed to be any function After choosing, is determined up to and Also: does not put constraints on : is allowed to be any function After choosing, is determined uniquely LINEAR DIFFERENTIAL SYSTEMS p14/22
47 Free variables Let, Let, The functions from only the components in the index set, form again a linear differential system (the elimination theorem, see lecture 3) Denote it by obtained by selecting LINEAR DIFFERENTIAL SYSTEMS p15/22
48 Free variables Let, Let, The functions from only the components in the index set, form again a linear differential system (the elimination theorem, see lecture 3) Denote it by obtained by selecting The set of variables is called free in if where, the cardinality of the set LINEAR DIFFERENTIAL SYSTEMS p15/22
49 Maximally free Let, Let The set of variables maximally free in such that we have if it is free, and if for any is called LINEAR DIFFERENTIAL SYSTEMS p16/22
50 Maximally free Let, Let The set of variables maximally free in such that we have if it is free, and if for any is called So: if is maximally free, then any set of variables obtained by adding to this set one or more of the remaining variables is no longer free LINEAR DIFFERENTIAL SYSTEMS p16/22
51 Input/output partition Let, Possibly after permutation of its components, a partition of into, with an input/output partition in free, is called if and is maximally LINEAR DIFFERENTIAL SYSTEMS p17/22
52 Input/output partition Let, Possibly after permutation of its components, a partition of into, with an input/output partition in free, is called if and is maximally In that case, is called an input of, and is called an output of Usually, we write for, and for LINEAR DIFFERENTIAL SYSTEMS p17/22
53 Input/output partition Let, Possibly after permutation of its components, a partition of, with an input/output partition in free if, is called and into is maximally In that case, is called an input of, and is called an output of Usually, we write for, and for Non-uniqueness: for given, the manifest variable in general allows more than one input/output partition LINEAR DIFFERENTIAL SYSTEMS p17/22
54 Input/output representations Let be the system with kernel representation where, and LINEAR DIFFERENTIAL SYSTEMS p18/22
55 Input/output representations Let be the system with kernel representation where, and Question: Under what conditions on the polynomial matrices and is an input/output partition (with input and output )? LINEAR DIFFERENTIAL SYSTEMS p18/22
56 Input/output representations Let be the system with kernel representation where, and Question: Under what conditions on the polynomial matrices and is an input/output partition (with input and output )? Proposition: is an input/output partition of with input, output if and only if is square and LINEAR DIFFERENTIAL SYSTEMS p18/22
57 Input/output representations Let be the system with kernel representation where, and Question: Under what conditions on the polynomial matrices and is an input/output partition (with input and output )? Proposition: is an input/output partition of with input, output if and only if is square and The representation input/output representation of is then called an The rational matrix is called the transfer matrix of wrt the given input/output partition LINEAR DIFFERENTIAL SYSTEMS p18/22
58 Does every have an input/output representation? LINEAR DIFFERENTIAL SYSTEMS p19/22
59 Does every have an input/output representation? Theorem: Let, with manifest variable There exists (possibly after permutation of the components) a componentwise partition of into, and polynomial matrices,,, such that LINEAR DIFFERENTIAL SYSTEMS p19/22
60 Does every have an input/output representation? Theorem: Let, with manifest variable There exists (possibly after permutation of the components) a componentwise partition of into, and polynomial matrices,,, such that There even exists such a partition such that is proper LINEAR DIFFERENTIAL SYSTEMS p19/22
61 Input and output cardinality has many input/output partitions However: the number of input components in any input/output partition of is fixed This number is denoted by and is called the input cardinality of : is free in LINEAR DIFFERENTIAL SYSTEMS p20/22
62 Input and output cardinality has many input/output partitions However: the number of input components in any input/output partition of is fixed This number is denoted by and is called the input cardinality of : is free in The output cardinality of, denoted by, is the number of output components in any input/output partition of Obviously: LINEAR DIFFERENTIAL SYSTEMS p20/22
63 Input and output cardinality has many input/output partitions However: the number of input components in any input/output partition of is fixed This number is denoted by and is called the input cardinality of : is free in The output cardinality of, denoted by, is the number of output components in any input/output partition of Obviously: For : LINEAR DIFFERENTIAL SYSTEMS p20/22
64 Summarizing Linear differential systems: those decribed by linear constant coefficient differential equations, etc LINEAR DIFFERENTIAL SYSTEMS p21/22
65 Summarizing Linear differential systems: those decribed by linear constant coefficient differential equations, etc Behavior := the set of solutions (for convenience) LINEAR DIFFERENTIAL SYSTEMS p21/22
66 Summarizing Linear differential systems: those decribed by linear constant coefficient differential equations, etc Behavior := the set of solutions (for convenience) polynomial matrix, representation of the system induced by a kernel LINEAR DIFFERENTIAL SYSTEMS p21/22
67 Summarizing Linear differential systems: those decribed by linear constant coefficient differential equations, etc Behavior := the set of solutions (for convenience) polynomial matrix, representation of the system induced by a kernel For a given system, is an input/output partition if the set of components of is maximally free: these components can be chosen arbitrarily The components of are then determined up to initial conditions LINEAR DIFFERENTIAL SYSTEMS p21/22
68 Summarizing Linear differential systems: those decribed by linear constant coefficient differential equations, etc Behavior := the set of solutions (for convenience) polynomial matrix, representation of the system induced by a kernel For a given system, is an input/output partition if the set of components of is maximally free: these components can be chosen arbitrarily The components of are then determined up to initial conditions An input/output representation of is a special kind of kernel representation:,, with partition Equivalent with: is an input/output LINEAR DIFFERENTIAL SYSTEMS p21/22
69 A system has in general many i/o representations, so also i/o partitions LINEAR DIFFERENTIAL SYSTEMS p22/22
70 A system has in general many i/o representations, so also i/o partitions Given, the number of components of is the same in any i/o partition This number is called the input cardinality of LINEAR DIFFERENTIAL SYSTEMS p22/22
71 End of LINEAR DIFFERENTIAL SYSTEMS p23/22
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