bangbang - lenhart
TRANSCRIPT
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Bang-bang and SingularControlsSuzanne Lenhart
University of Tennessee, Knoxville
Departments of Mathematics
Supported by NIH grantLecture4 – p.1/ ??
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Bang-Bang Control
A solution to a problem that is linear in thecontrol frequently involves discontinuities in the
optimal control
Lecture4 – p.2/ ??
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Contd.
rest
where is switching function. Maximize w.r.t. at
if
if
if
Lecture4 – p.3/ ??
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If
is not sustained over aninterval of time, then the control is bang-bang.
Bang-bang always at the extreme values of thecontrol set.
a
b
t
u*
Lecture4 – p.4/ ??
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Example
max sales
stock can be reinvested to expand capacityor sold for revenue
fraction of stock to be reinvested
Lecture4 – p.6/ ??
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when
when
is decreasing and
.
On
.
Lecture4 – p.7/ ??
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Switch time
If
If
Lecture4 – p.8/ ??
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Example
no transversality condition
Lecture4 – p.9/ ??
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is switching function and it willswitch from to at most once.at
on
If on
DE
That solution does not satisfy .Lecture4 – p.10/ ??
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There must be one switch.On
DE
must be continuous at
Switching function at
Lecture4 – p.11/ ??
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Lecture4 – p.12/ ??
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Basic resource Model
(Clark p. 95) “Fishery”
control (effort), price and “catchability”
max discounted protrevenue - cost
Lecture4 – p.13/ ??
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Singular control occurs when the coefcient ofcontrol is zero over a time interval (switchingfunction is zero).
when
singular case
Lecture4 – p.14/ ??
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Suppose
on a time interval. Solve for
Lecture4 – p.15/ ??
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From necessary conditions
after substituting in
. Set 2 expressions for
equal
cancel
Lecture4 – p.16/ ??
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Term involving control cancel
...
optimal state should satisfy this equation when inthe singular control case.(nd and use state DE to nd ).
Lecture4 – p.17/ ??
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Simple case
ignoring cost of shing
Singular case
during singular case
Lecture4 – p.18/ ??
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During singular case
If
, singular case
Lecture4 – p.19/ ??
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Back to Bioreactor Model
state (bacteria)
control (nutrient input)
Lecture4 – p.20/ ??
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If
on a time interval, singular control may satisfy
using
DE
Lecture4 – p.21/ ??
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is a strictly decreasing function. Thus
cannot be maintained on an interval. No singularcase here.One switch occurs when
can show if and sufciently large, there
is a switch.Lecture4 – p.22/ ??
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If
or large
Where is decay rate of contaminant and isgrowth rate of bacteriaand sufciently large, there is a switch
tT
M
u*
T̂
Otherwise i.e. no nutrient feeding.
Lecture4 – p.23/ ??