Fundamental Theorem of Calculus when Non-Conservative
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Let's say I have an integral of an objective function which is path-dependent (such as in an optimal control problem):
$$ J = int_{t_0}^{t_f} lbig( x(t) big) dt$$
And want to compute the following gradient of $J$ with respect to $x(t)$ at the time $t_f$, or $frac{dL}{dx(t)} rvert_{x(t_f)}$. Can I still use the fundamental theorem of calculus such that:
$$frac{dL}{dx(t)} rvert_{x(t_f)} = lbig( x(t_f) big)$$
If not, then how would I be able to find the above gradient?
calculus line-integrals
asked Dec 19 '18 at 4:40
Jacob SacksJacob Sacks
11
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add a comment |
$begingroup$
Let's say I have an integral of an objective function which is path-dependent (such as in an optimal control problem):
$$ J = int_{t_0}^{t_f} lbig( x(t) big) dt$$
And want to compute the following gradient of $J$ with respect to $x(t)$ at the time $t_f$, or $frac{dL}{dx(t)} rvert_{x(t_f)}$. Can I still use the fundamental theorem of calculus such that:
$$frac{dL}{dx(t)} rvert_{x(t_f)} = lbig( x(t_f) big)$$
If not, then how would I be able to find the above gradient?
calculus line-integrals
asked Dec 19 '18 at 4:40
Jacob SacksJacob Sacks
11
$endgroup$
add a comment |
$begingroup$
Let's say I have an integral of an objective function which is path-dependent (such as in an optimal control problem):
$$ J = int_{t_0}^{t_f} lbig( x(t) big) dt$$
And want to compute the following gradient of $J$ with respect to $x(t)$ at the time $t_f$, or $frac{dL}{dx(t)} rvert_{x(t_f)}$. Can I still use the fundamental theorem of calculus such that:
$$frac{dL}{dx(t)} rvert_{x(t_f)} = lbig( x(t_f) big)$$
If not, then how would I be able to find the above gradient?
calculus line-integrals
asked Dec 19 '18 at 4:40
Jacob SacksJacob Sacks
11
$endgroup$
Let's say I have an integral of an objective function which is path-dependent (such as in an optimal control problem):
$$ J = int_{t_0}^{t_f} lbig( x(t) big) dt$$
And want to compute the following gradient of $J$ with respect to $x(t)$ at the time $t_f$, or $frac{dL}{dx(t)} rvert_{x(t_f)}$. Can I still use the fundamental theorem of calculus such that:
$$frac{dL}{dx(t)} rvert_{x(t_f)} = lbig( x(t_f) big)$$
If not, then how would I be able to find the above gradient?
calculus line-integrals
calculus line-integrals
asked Dec 19 '18 at 4:40
Jacob SacksJacob Sacks
11
asked Dec 19 '18 at 4:40
Jacob SacksJacob Sacks
11
asked Dec 19 '18 at 4:40
Jacob SacksJacob Sacks
11
asked Dec 19 '18 at 4:40
Jacob SacksJacob Sacks
11
asked Dec 19 '18 at 4:40
Jacob SacksJacob Sacks
11
11
add a comment |
add a comment |
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